Maximizing minimal rights for sustainability: a viability approach

Abstract

This paper examines how the viability approach can be used to define sustainability goals. In an economic model with a non renewable natural resource, we define minimal rights to be guaranteed for all generations. These rights can include a minimal consumption (economic goal) and the preservation of natural resources (environmental goal). From a given economic state, it is possible to define the set of minimal rights that can be provided for all generation. To address the intergenerational equity issue, we propose to use a criterion that define the set of minimal rights that provide the maximal utility, in a Rawlsian perspective (Rawls, 1971). We describe how this criterion can be applied and computed, and discuss it with respect to usual criteria, including the maximin criterion, the Green Golden Rule, the Chichilnisky approach and the Mixed Bentham-Rawls criterion.

Full text

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Maximizing minimal rights for sustainability: a
viability approach
V. Martinet∗
March 21, 2007
Abstract
This paper examines how the viability approach can be used to define
sustainability goals. In an economic model with a non renewable natural re-
source, we define minimal rights to be guaranteed for all generations. These
rights can include a minimal consumption (economic goal) and the preser-
vation of natural resources (environmental goal). From a given economic
state, it is possible to define the set of minimal rights that can be provided
for all generation. To address the intergenerational equity issue, we pro-
pose to use a criterion that define the set of minimal rights that provide the
maximal utility, in a Rawlsian perspective (Rawls, 1971). We describe how
this criterion can be applied and computed, and discuss it with respect to
usual criteria, including the maximin criterion, the Green Golden Rule, the
Chichilnisky approach and the Mixed Bentham-Rawls criterion.
Key-words: sustainability, intergenerational equity, minimal rights, viability.
JEL Classification: Q01, Q32, O13, C61.
∗EconomiX, Universit´ e Paris X Nanterre, 200 av. de la R´ epublique, 92001 Nanterre Cedex.
Email: vincent.martinet@u-paris10.fr
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Contents
1Introduction3
2On the possibility to achieve some minimal rights
2.1Viability framework and minimal rights . . . . . . . . . . . . . . . .
2.2Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Admissible sustainability objectives for a given initial state . . . . .
7
7
8
9
3 Choosing among sustainability goals
3.1 An objective function related to minimal rights
3.2Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
11
12
15
. . . . . . . . . . .
4 Discussion
4.1Minimal rights and sustainability criteria . . . . . . . . . . . . . . .
4.2Defining minimal objectives to implement sustainability . . . . . . .
4.3 Reaching minimal right and minimal time of crisis . . . . . . . . . .
16
16
20
21
5 Conclusion22
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1Introduction
Solow (1993, p.167-168) claimed that
“If the sustainability means anything more than a vague emotional
commitment, it must require that something be conserved for the very
long run. It is very important to understand what that thing is: I think
it has to be a generalized capacity to produce economic well-being.”
Asheim et al. (2001) argue that sustainability requires the utility to be non
decreasing along time, for equity concerns. Stavins et al. (2003) interpret sustain-
ability as ‘intergenerational equity’ plus ‘economic efficiency’. Intergenerational
equity is defined by requiring a non decreasing utility through time, and economic
efficiency is defined with respect to the neo-classical discounted utilitarian crite-
rion. In these approaches, the social objective (sustainability of the utility) is not
considered in the objective function (in the criterion to optimize) but as an added
constraint to an economic criterion. This approach is criticized by Krautkraemer
(1998), Pezzey and Toman (2002) and Cairns and Long (2006) who argue that the
objective function has to be defined in order to consider the sustainability issue,
and especially intergenerational equity. From that point of view, the debate on
sustainability becomes a debate on the social criterion to be optimized.
Heal (1998) examines the various criteria proposed to cope with the sustainabil-
ity issue. Each criterion characterizes the optimal (sustainable) path and defines
what is preserved for sustainability (if anything is preserved).
The most commonly used criterion is the intertemporal sum of discounted
utilities
max
0
This criterion, which has a solution only if the discount factor ∆(t) decreases to-
wards zero at the infinite time1, is criticized because it does not take long term
utility into account. According to Chichilnisky (1996) this criterion is a dictator-
ship of the present.
?∞
∆(t)Utdt.(1)
Another proposed criterion (Beltratti et al., 1995) is inspired by the economic
Golden Rule, and defines the highest indefinitely maintainable level of instanta-
neous utility. The criterion reads
max lim
t→∞Ut
(2)
1Usually, the discount factor is decreasing at a constant rate, i.e.
alternatives have been proposed, including hyperbolic discounting which lead to a decreasing
discount rate. A general form of hyperbolic discounting is ∆(t) = (1 + αt)−γ/α. Nevertheless,
the discount factor is still decreasing and the utility of the far future is not taken into account,
which raises the equity issue.
∆(t) = e−δt.Other
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This approach, called the green golden rule, does not take into account the present,
and is called a “dictatorship of the future” by Chichilnisky (1996).
Solow (1974) and Cairns and Long (2006) address the sustainability issue by
using the maximin criterion. This criterion defines the maximal sustainable level
of utility, in the sense that it maximizes the utility of the poorest generation:
?
This criterion has been criticized as it may lead to maintain the initial poverty by
restricting the investment if the first generation is the poorest.
maxmin
t
Ut
?
(3)
As mentioned by Beltratti et al. (1995, p.179), “an important task ...lies in the
analysis of criteria that combine maximization of discounted utility with elements
related to the long run”. In other words, sustainability issue relies on the definition
of criteria that take into account both short and long run. Chichilnisky (1996)
develops such a study and provides the general form that the criterion must have:
?
where ∆(t) is the discount factor. Nevertheless, the solution of criteria of this form
is not easy to compute (See Heal, 1998). Moreover, the criterion is not unique and
depends on the choice of the parameter θ and of the discount factor ∆(t).2
maxθ
?∞
0
∆(t)Utdt + (1 − θ) lim
t→∞Ut
?
(4)
In a recent contribution, Long (2006) proposes, in a complementary way of
Chichilnisky (1996), to mix a maximin criterion and the utilitarian discounted
criterion. The criterion he proposes, called a “Mixed Bentham-Rawls criterion”
reads3
?
This criterion is sensitive to the present, to the future and to the least advantaged
generation.
maxθ
?∞
0
∆(t)Utdt + (1 − θ)min
t
Ut
?
.(5)
As mentioned in the begining of this introduction, sustainability can be in-
terpreted as the requirement to have something preserved in the long-run, in an
intergenerational equity perspective. To discuss various criteria with respect to
their implications4, the debate mainly emerges in the intertemporal resource al-
location framework and aims at defining an equitable share of consumption and
2The term lim can also be replaced by any function that depends only on the limiting behavior
of the utility over time, such as long-run average for example.
3Long (2006) develops the criterion in a discreet time framework and over a finite time horizon.
He also consider a constant discount rate. To be consistent within the presentation of the various
criteria, we present the criterion a continuous time over an infinite horizon.
4According to Dasgupta and Heal (1979, p.311), “it is legitimate to revise or criticize ethical
norms in the light of their implications”.
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access to natural resources among generations. Criteria are more or less preserva-
tive with respect to the resource use and the level of consumption. The maximin
criterion is the most explicit in the definition of what is preserved, as it targets
the preservation of the utility along time.
In particular, the equity issue is mainly addressed with respect to the con-
sumption of each generation, and the utility function depends on the instanta-
neous consumption ct, namely U(ct). Each criterion leads to a infinite stream of
consumption. The emergence of environmental issue has lead to to into account
natural resource amenities, which emphasizes the preservation issue. In that per-
spective, the utilitarian approach can be completed: Natural resources xtcan be a
component of the utility function, U(ct,xt). Heal (1998) examines the implications
of such an approach in the sustainability debate. Depending on the criteria, there
is more or less consumption and thus “less or more” preservation of resource stocks.
The Brundtland report, Our Common Future (WCED, 1987), defined the sus-
tainable development as a “development that meets the needs of the present with-
out compromising the ability of future generations to meet their own needs”. This
explicit reference to the “needs” of the various generations can be a way to address
the issue. An important challenging question is then the definition of the basic
needs to be guaranteed for a sustainable development.
In the late 70’s, the economic development issues where much more concerned
with the intragenerational equity issue, and the development of poor countries.
It has been argued (Chichilnisky, 1977) that aggregated economic indicators like
GDP were not able to encompass the development issue, as it neglects an important
dimension of development: the satisfaction of basic needs. In fact, the scarcity
of some basic goods can jeopardize economic development, or at least, modify
the priority of the social planner: the maximization of Present Net Value of the
economy may not be the primary goal of development.
From a more general point of view, one way to address the sustainability issue
is to consider the satisfaction of basic needs, or equivalently, of minimal rights.
Among these basics needs, one can consider a minimal consumption and a mini-
mal environmental quality. The issue is then to determine the way we define these
minimal rights, in an equity concern. According to John Rawls’ conception of
justice (Rawls, 1971), the first requirement for equity is to choose the allocation
of resources that provides the maximal number of minimal rights every one can
enjoy.5Even if the Rawls’ conception of justice was not built for intergenerational
5This result comes from the allocation of rights one would made under the “veil of ignorance”.
Rawls argues that justice should be based on two principles, with a priority order. The first
principle is the definition of fundamental rights every one can enjoy (“each person is to have an
equal right to the most extensive scheme of equal basic liberties compatible with a similar scheme
of liberties for others”). The second principle is based on (with here again a priority order) “fair
equality of opportunity” to a social position and on the “difference principle” that stipulates that
inequality in the wealth distribution is justified if it is beneficial for the poorest individual, i.e.
if the poorest individual in this configuration in richer than the poorest individual in all other
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equity issues6, we can examine what minimal rights can be provide to all genera-
tions. In our sustainable development issue, it is equivalent to examine what level
of consumption, and what level of environmental quality can be provided to all
generations.
In a way similar to Chichilnisky (1977) that argues that economic development
must be consistent with the attainment of adequate levels of per capita consump-
tion of basic goods, a primary goal to achieve intergenerational equity can be define
as a minimal per generation access to basic goods. The satisfaction of basic needs
criterion aims at guaranteeing that each generation gets a minimal level of pri-
mary goods, while allowing some generation to have more than what is considered
sufficient.
In this paper, we propose a methodological approach to determine minimal
rights representing sustainability. Minimal rights are defined as a set of constraints
(sustainability objectives) that must be respected forever, in an intergenerational
equity perspective. Such constraints refer to basic needs of each generation. Chat-
terjee and Ravikumar (1999) study the implications of minimum consumption
requirements on the rate of growth and the evolution of wealth distribution. We
focus here on the implications of such a constraint on intertemporal consumption
and investment paths, and on intergenerational equity, when the economy relies
on the extraction of a non-renewable resource. At the same time, we consider
a minimal resource constraint ensuring the preservation of a part of the natural
resource.
Using the viability framework, we exhibit intertemporal paths that make it
possible to satisfy a set of constraints representing minimal rights for all genera-
tions, including a minimal consumption level and a guaranteed resource stock to
be preserved. It is a much more general way to address the minimal consumption
and viability issue than found in Dawid and Day (2006).
Our approach is however quite different from the sufficientarianism. We do not
take a given level of minimal consumption and resource preservation, and then
determine the intertemporal path that make it possible to satisfy these particular
constraints in the future, like in Chichilnisky (1977). On the contrary, we aim at
defining the set of all the achievable goals from the initial state, i.e. the set of
minimal consumption levels and preserved resource stock that can be guaranteed
forever given the initial state. We then propose a criterion to choose within the
set of all possible sustainability goals.
It leads to define a possible mathematical formalization to define minimal rights
to be provided for each generation. In that framework, we do not optimize in-
tertemporal paths but level of constraints to be guaranteed forever.
possible allocations. This last statement leads to the maximin criterion, which is thus the less
important point in Rawls’ theory of justice.
6According to Rawls, his theory of justice should not be applied to intergenerational problems.
Nevertheless, one of the propositions of Rawls (1971) has already been applied to intertemporal
equity problems: the maximin criteria (Solow, 1974; Cairns and Long, 2006).
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This approach is applied to an intertemporal resource allocation model with a
manufactured capital stock and a non renewable natural resource. It is a canonical
model severally used in the literature on sustainability (Dasgupta and Heal, 1974;
Solow, 1974; Heal, 1998), which allow a comparison of results with the existing
criteria, including the maximin, the green golden rule, the Chichilnisky approach
and the mixed Bentham-Rawls criterion.
The paper is organized as follows. In section 2, we define the set of minimal
rights that can be guaranteed to all generations in a standard economic model with
an exhaustible resource. We present the framework of analysis (2.1), the model
(2.2), and the results (2.3). A criterion based on the maximization of minimal
rights is proposed in 3, and applied to our analytical model. The results are
interpreted in section 4. We discuss the implications of this criterion with respect
to the results of other criteria (maximin, green golden rule, Chichilnisky’s criterion,
mixed Bentham-Rawls criterion). Section 5 concludes.
2 On the possibility to achieve some minimal
rights
In this section, we develop a framework to define the set of sustainability goals
that can be achieved by an economy, given initial endowments. This framework is
based on an extension of the viable control framework (Aubin, 1991).
2.1 Viability framework and minimal rights
If we consider that sustainability encompasses several objectives from different na-
ture (ecological, economic and social) it may be difficult to adopt an optimization
approach and define efficiency in a multicriteria context. Another possibility is
to adopt the viability approach, which studies the effectiveness of dynamic in-
tertemporal paths, with respect to a set of objectives defined by constraints. In
the viability framework, the constraint must be respected forever, inducing an
intergenerational equity perspective. The constraints can thus be interpreted as
minimal rights to be guaranteed to any generation.
The purpose of the viability approach (Aubin, 1991) is to study the behavior
of dynamic systems subject to constraints. The desired characteristics for the
system are defined by a set of constraints, and the viability approach examines the
conditions for these constraints to be satisfied forever, in a dynamic perspective.
The first step of the analysis is to define the states of the system from which at
least one intertemporal path that satisfies the constraints forever starts. The set
of such states is called the viability kernel of the problem. By definition, from any
state in the viability kernel, there is at least one admissible decision that keeps
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the trajectory in the kernel. Thus, from any state outside the viability kernel,
there are no decisions that make it possible to respect the constraints forever.
Whatever the decisions, the dynamics of the system will violate the constraint in
a finite time.7Viable paths are paths that stay in the kernel. The second step of
the analysis is the definition of the viable decisions associated with a viable state.
Not all decisions may be viable and some of the admissible decisions may lead the
system outside the viability kernel.
It means that if the initial state of the economy is not within the viability ker-
nel, the initial endowments of the economy do not allow the sustainability goals
to be achieved. The objectives (constraints level) should be diminished to be ful-
filled. In that context, it is possible to extend the viability analysis by defining,
for a given initial state, the set of reachable goals.
In our context, the constraints include a minimal consumption c and a minimal
resource stock x to be preserved. We will examine the set of goals (c,x) that are
reachable from the initial capital stocks of the economy. Doing that, we define the
set of minimal rights that can be guaranteed to any generation.
As mentioned in the introduction, a particular problem one can face when try-
ing to achieve some particular goals is that the defined goals may not be reachable
from initial conditions through feasible paths of the system. In that case, the goals
can be reached in some future date. This issue is discussed in section 4.3.
2.2 Model
To make comparisons with other approaches easier, we develop the proposed ap-
proach for a standard model. We consider an economy with infinitely many gen-
erations, and make the simplifying assumption that each generation is composed
by an unique representative agent.8
A non renewable resource xt is extracted
(rt) and used with capital ktto produce capital. The production function is de-
noted f(kt,rt). The capital can be ever consumed or invested. Such a model has
been studied in Dasgupta and Heal (1974); Solow (1974), and is an useful styl-
ized model for addressing the sustainability issue: the intertemporal allocation
of the exhaustible resource, and the stream of consumption through time make
intertemporal comparisons possible, in an intergenerational equity perspective.
The dynamics are
˙k = kα
˙ x = −rt.
trβ
t− ct, (6)
(7)
7This is related to the inertia of the system.
constraints but from which no viable path exists. The crisis is unavoidable.
8The intragenerational equity issue is not addressed. However, the perspective of growing
population is discussed in section 4.3, in a discussion on the possibility to increase minimal
rights along time.
There are states that satisfy the viability
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We take a Cobb-Douglas production function which is argued to be the most
interesting case for studying sustainability, as mentioned by Dasgupta and Heal
(1979).9
In that model, we wonder what are the level of guaranteed consumption c and
preserved resource stock x that are compatible with the initial state (k0,x0). We
are thus considering the following viability constraints
ct ≥ c,
xt ≥ x.
(8)
(9)
2.3Admissible sustainability objectives for a given initial
state
In order to define the minimal rights to be guaranteed for a sustainable devel-
opment, we need to define the set of all possible rights, i.e. the rights that are
compatible with the initial state of the economy and with the studied dynamics.
We want to define the set of
S =
?
(c,x)
????
there exist paths (k(.),x(.)) starting from (k0,x0)
that satisfy the constraints (8) and (9)
?
. (10)
Martinet and Doyen (2007) describe the relationship between consumption and
preservation goals in the model we consider. They examine the conditions for
a minimal consumption c to be guaranteed when there is also a constraint on
the preservation of the resource x. It means that they determine if there are
intertemporal decisions (c(.),r(.)) that make it possible to respect the constraints
ct≥ c and xt≥ x for all t ≥ 0, from a given initial state (k0,x0).
As a consequences it is possible to determine which levels of guaranteed con-
sumption c and preserved resource stock x are reachable from the initial state.
An easy way to define levels of consumption that can be guaranteed along with
the preservation of a part of the resource stock is to consider the maximal level of
consumption that can be sustained, under a resource preservation constraint.
?
such that ∀t ≥ 0,ct≥ c?and xt≥ x
c+(k0,x0,x) = maxc???given (k0,x0),there exists (c(.),r(.))
?
.(11)
9If we consider Constant Elasticity of Substitution (CES) production functions, if the elasticity
of substitution between the capital and the resource is greater than one, there is no sustainability
issue as it is possible to produce without using the natural resource. The resource is not essential
to produce. On the contrary, an elasticity of substitution lower than one implies that the resource
is essential to produce, and the intertemporal production is bounded. No consumption can be
sustained. The intermediate case with elasticity equals to one, i.e. the Cobb-Douglas case, makes
it possible to substitute capital to the resource in the production even if the resource is essential
to produce. According to Dasgupta and Heal (1979), this case is the most interesting.
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This approach is linked with the maximin approach (Solow, 1974; Cairns and
Long, 2006). In the classical model we study, the maximal sustainable level of
consumption, given the preservation goal x, is defined as follows.
Result 1 Consider a Cobb-Douglas function with α > β. The maximal sustain-
able consumption from initial state (k0,x0), associated with the guaranteed stock x
is
c+(k0,x0,x) = (1 − β)?(x0− x)(α − β)?
This result is proven in Martinet and Doyen (2007, Proposition 3), but an easier
way to insure this result is to note that equation (12) leads exactly to the maximin
result of Solow (1974, p.39) for x = 0. Solow’s result is thus extended to take into
account the resource constraint.
Fig. 1 represents equation (12). It represents all consumption and resource
preservation levels that can be guaranteed to all generation. Eq. (12) is the upper
bound of the set of all reachable goals.
β
1−βk
α−β
1−β
0
. (12)
02468 1012 14 16
0. 1
0.3
0.7
1.1
1.5
1.9
2.3
0.0
Xo
max(c+)
c+( x )
x
Figure 1: Substitution between guaranteed consumption and resource conservation
for a Cobb-Douglas technology.
We thus know the set of all reachable goals for sustainability. Any inner pair
(c,x) such that c ≤ c+(x) is feasible.10Note that on the border, a rise of resource
preservation implies a fall of sustainable consumption.
A question that arises now is to define sustainability goals in order to satisfy
some intergenerational equity.
10For the sake of simplicity, we will omit the initial state in the notation of function
c+(k0,x0,x).
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3Choosing among sustainability goals
In this section we introduce an objective function to define which sustainability
rights should be chosen among the possible set S.
3.1An objective function related to minimal rights
Consider that minimal rights are based on a minimal consumption c and a minimal
resource stock x. We define social preferences among the minimal rights to be
guaranteed.
Definition 1 A minimal right utility function U(c,x) represents the preferences
on minimal rights. We postulate
• c ∈ R+
• U(c,x) : R+
• Uc≥ 0 ; Ux≥ 0
• Uc,x≤ 0.
This utility function represents the preferences of a virtual representative agent
placed under the veil of ignorance, and that should choose minimal rights to be
guaranteed to any generation in a sustainability concern.
0; x ∈ R+
0× R+
0
0?→ R+
0
We propose to choose the pair of minimal rights (c,x) that maximizes this
utility function. Hence, our approach is not to maximize some intertemporal
welfare but to maximize the utility associated with some minimal rights to be
guaranteed for all generations.
Definition 2 We define the Minimal Rights criterion as
max
c,xU(c,x) (13)
subject to
(k0,x0)given (14)
(15)
(16)
(17)
(18)
ct ≥ c
xt ≥ x
˙k = kα
˙ x = −rt
trβ
t− ct
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3.2Results
Using the result 1, this problem is equivalent to the maximization of this criterion
among the possible pairs (c,x), i.e.
max
c,xU(c,x)
s.t.
≤
≤
≤
≤
0
0
0
0
x
c
x0− x
c+(x) − c
(19)
(20)
(21)
(22)
This problem is a classical static optimization problem under inequality con-
straints (L´ eonard and Long, 1992).11
To solve this problem, we define the following functional form
φ(µ1,µ2,µ3,µ4,c,x) = U(c,x) + µ1x + µ2c + µ3(x0− x) + µ4(c+(x) − c)
where the µiare the dual variables of the problem.
According to the Khun-Tucker theorem, the optimality conditions of the prob-
lem are12
(23)
φµ1= x ≥ 0,
φµ2= c ≥ 0,
φµ3= x0− x ≥ 0,
φµ4= c+(x) − c ≥ 0,
φx= Ux+ µ1+ µ4dc+(x)
µ1≥ 0,
µ2≥ 0,
µ3≥ 0,
µ4≥ 0,
x ≥ 0,
c ≥ 0,
µ1x = 0
µ2c = 0
µ3(x0− x) = 0
µ4(c+(x) − c) = 0
x
(24)
(25)
(26)
(27)
dx
≤ 0,
?
Ux+ µ1+ µ4dc+(x)
dx
?
= 0 (28)
φc= Uc+ µ2− µ4≤ 0,c(Uc+ µ2− µ4) = 0 (29)
Strictly positive solutions
First assume that, at the optimum, both the optimization variables x and c are
strictly positives. From eq. (24) and (25), we get µ1 = µ2 = 0. Moreover, if
consumption is positive, the preserved resource stock will be lower than the initial
stock x0. Eq. (26) then leads to µ3= 0. We thus get a system from equations
(27), (28) and (29), in which there are three equations and three variables.
µ4(c+(x) − c) = 0
x
(30)
?
Ux+ µ4dc+(x)
dx
?
= 0 (31)
c(Uc− µ4) = 0 (32)
11Such problems are generally easier to solve than complex dynamic optimization programs.
12The variables are at optimal values. In order to get simple notations, we do not note them
x∗and c∗but simply x and c
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As we have assumed that c ?= 0, eq. (32) leads to µ4= Uc. We thus get from
eq. (30) and (31) the conditions
c = c+(x)
dc+(x)
dx
(33)
= −Ux
Uc
(34)
It leads to the following result
dc+(x)
dx
= −
Ux
Uc+(x)
(35)
Fig. 2 illustrates this result.

c+(k0,x0,x)
U(c,x)
x0
c
x
Figure 2: Optimal minimal rights when c > 0 and x > 0.
Corner solution x = 0
Assume now that x = 0. It implies that µ3= 0 (from eq. 26).
If c = 0, eq. (27) would require µ4= 0. But it is in contradiction with relation
(29) which requires Uc+ µ2− µ4≤ 0. Thus, we have µ4> 0 and c = c+(0), from
eq. (27).
As c ?= 0, we get µ2= 0 from eq. (25). We then get µ4= Ucfrom eq.(29).
Finally, the inequality condition (28) requires
Ux|(x=0)+ µ1+ Ucdc+(x)
dx
|(x=0)
≤ 0 (36)
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This equation can be expressed with respect to µ1
µ1≤ −Ux|(x=0)− Ucdc+(x)
dx
|(x=0)
(37)
As µ1≥ 0, it is possible only if −Ux|(x=0)− Uc
dc+(x)
dx
|(x=0)≥ 0, or equivalently if
0 ≤Ux
Uc
≤ −dc+(x)
dx
|(x=0)
(38)
It means that such a corner solution is possible if the slope of the utility function
is smaller than that of the function c+(x). Such a solution is then possible only if
the marginal utility of preservation for a nil resource stock is small with respect
to the marginal utility of consumption. It implies that Ux|(x=0)< ∞.
Fig. 3 illustrates this result.

c+(k0,x0,x)
U(c,x)
x0
c
x
Figure 3: Optimal minimal rights when x = 0.
Corner solution c = 0
We now turn toward the other case : c = 0. The inequality from eq. (29) implies
Uc+ µ2− µ4≤ 0=⇒
µ4≥ Uc|(c=0)+ µ2> 0 (39)
It is only possible if Uc|(c=0)< ∞.
As µ4> 0, we know from eq. (27) that c = c+(x) which means, as c = 0, that
x = x0. We have µ1= 0 from eq. (24). Thus, x > 0 requires from eq. (28) that
µ4= −
Ux|(x=x0)
dc+(x)
dx
|(x=x0)
(40)
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Combining this condition with eq. (39), we get
0 ≤ µ2≤ −Uc|(c=0)−
Ux|(x=x0)
dc+(x)
dx
|(x=x0)
(41)
We thus have a condition on the marginal utilities in (c = 0,x = x0):
0 ≤ −dc+(x)
dx
≤Ux
Uc
(42)
The slope of the utility function must be greater than that of the function c+(x).
In particular, the marginal utility of consumption, when consumption is zero, must
be finite (and small with respect to the marginal utility of preservation).
Fig. 4 illustrates this result.

c+(k0,x0,x)
U(c,x)
x0
c
x
Figure 4: Optimal minimal rights when c = 0.
3.3 Interpretation
This approach has methodological intersections with the maximin approach (Solow,
1974; Cairns and Long, 2006), in that the efficient growth paths are solutions of a
constrained optimization problem: maximin under constraint
Whatever the case, the optimal solution always satisfies c = c+(x). The only
intertemporal path that satisfies the optimal minimal rights is then a maximin un-
der constraints and is efficient from an economic point of view. As a consequence,
optimal minimal rights with respect to our criterion entail efficient resource use.
The only intertemporal path that satisfies the optimal constraint is the same path
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