invariance in growth theory and sustainable development

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Journal of Economic Dynamics & Control 31 (2007) 2827–2846

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Invariance in growth theory and sustainabledevelopment

Vincent Martinet�, Gilles Rotillon

EconomiX (UMR 7166 CNRS-UPX), Universite Paris X-Nanterre, 200 av. de la republique, 92001

Nanterre Cedex, France

Received 14 April 2005; accepted 4 October 2006

Available online 4 December 2006

Abstract

This paper analyzes the general concept of sustainability from a different point of view than

that generally found in the literature. If sustainability is defined as the requirement to keep

something constant or at least non-decreasing throughout time, the choice of the thing to be

preserved is controversial. Neo-classical models mainly assume that sustainability requires

that consumption or a utility level has to be preserved. In this paper, we object to this a priori

conception of sustainability and define all the quantities that can be preserved in neo-classical

optimal growth models. We thus wonder if invariant quantities can be found along the

optimal paths defined by a classical representation of an economy with an exhaustible

resource. We use the Noether theorem to determine the conservation laws of dynamic systems.

We examine under which conditions there is such invariance and how it could be interpreted as

a sustainability indicator. We emphasize the limits of the economic growth theory for coping

with the sustainability issue.

r 2006 Elsevier B.V. All rights reserved.

JEL classification: Q01; O40

Keywords: Sustainable development; Optimal control; Economic conservation laws; Hartwick rule

see front matter r 2006 Elsevier B.V. All rights reserved.

.jedc.2006.10.001

nding author.

dress: [email protected] (V. Martinet).

www.elsevier.com/locate/jedc

dx.doi.org/10.1016/j.jedc.2006.10.001

mailto:[email protected]

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V. Martinet, G. Rotillon / Journal of Economic Dynamics & Control 31 (2007) 2827–28462828

1. Sustainability as the conservation of something throughout time

In this paper, we propose a novel approach to sustainability. We search forendogenous invariant quantities in optimization problems. We then ask whethersuch constant quantities can be interpreted as ‘the thing to conserve forsustainability’. For this purpose, we examine and interpret the conditions underwhich these quantities remain constant. Let us detail our research program.

In the economic literature, the sustainable development issue is often tackled in theway: ‘Something must be kept constant, or at least not decreasing’, and the debate isabout the ‘thing’ to be preserved.1 Solow (1993, pp. 167–168) claimed that

If the sustainability means anything more than a vague emotional commitment, itmust require that something be conserved for the very long run. It is veryimportant to understand what that thing is: I think it has to be a generalizedcapacity to produce economic well-being.

Sustainability criteria can be used to determine what is to be conserved forsustainability (Heal, 1998). The most commonly used criterion is the neo-classicaldiscounted utility

max

Z 10

e�dtUðtÞdt.

This criterion is criticized mainly because the discount factor is decreasing2 and thecriterion does not take long-term utility into account. According to Chichilnisky(1996) this criterion is a dictatorship of the present.

An equity requirement is often added to the criterion. Asheim et al. (2001) andStavins et al. (2003) propose to require a non-decreasing utility or consumption level.The social objective (sustainability of the utility) is not considered in the objectivefunction (in the criterion to optimize) but as an added constraint to an economiccriterion. This approach is criticized by Krautkraemer (1998) and Cairns and Long(2006) who argue that the objective function has to be defined in order to considerthe sustainability issue, and especially intergenerational equity.

We agree with this point of view. For us, this approach is not relevant because itdefines sustainability as an a priori constraint, leading to the following steps: acriterion is chosen (it is often the discounted utility maximization). Then, theconstraints representing sustainability (constant consumption, non-decreasingutility...) are defined a priori. Finally, optimal paths are characterized. These pathsare then interpreted thanks to the a priori definition of sustainability, leading torecommendations of the form: ‘The economy follows a sustainable path if3 the utilityfunction (the production function, the pollution abatement function) is of theform...’. We reject this approach because the results are limited: how restrictions onthe form of the utility function can be justified? Does the social planner have to

1See Dobson (1996) for a typology of sustainabilities.2This discount factor can be hyperbolic.3One should read: ‘the program defined as representing sustainability has a solution if...’.

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V. Martinet, G. Rotillon / Journal of Economic Dynamics & Control 31 (2007) 2827–2846 2829

impose its form to the representative agent? We do not think that an ad hoc utilityfunction has to be chosen for the criterion to have a solution, or, roughly speaking,in order to get the solution one wants. Moreover, the hypotheses are strong: how canwhat has to be conserved for future generations be chosen a priori? The limit of thisapproach is that sustainability is considered both as a technical definition and amoral injunction. A definition that characterizes a particular development path astechnically feasible (for instance having a non-decreasing consumption, or thepossibility to preserve a natural resource stock) does not imply any moral strength tofollow it. With such an approach, it comes out that positive propositions concerningthe danger of a particular economical/environmental path cannot be separated fromthe possible optimality of such a path.

We also start from the idea that, if sustainability means anything, it requires to‘preserve something in the long-term’, but we do not characterize that thing a priori.On the contrary, our approach consists in wondering if there are invariant quantitiesendogenous to the representation of the economy. Such invariants will give asignificance to the ‘thing’ we can and perhaps we want to preserve. We thus wonderwhat it is possible to sustain in a production–consumption economy. We adopt ageneral approach to find criterions of the form

max

Z 10

ZðtÞUðtÞdt

that lead to such invariant quantities. It is the general formulation of a program thatmaximizes some intertemporal sum of utilities. ZðtÞ is a weightening function of anyform (not necessarily of the usual exponential form).

To summarize, we examine if the optimization of an utilitarian criterion makes itpossible to define invariant quantities that could be interpreted as sustainability. Wedo not define what has to be preserved, and thus we make no explicit assumptionsabout the definition of sustainability. We consider the conditions which allow anessential neo-classical aggregated economy to sustain something. We especiallyexamine conditions on the discount factor and technical change. For this purpose,we use the Noether theorem (Noether, 1918) to determine if optimal controlproblems – with utilitarian criteria and an exhaustible resource – lead to someinvariant quantities along optimal paths. This theorem relates symmetries indynamic optimization problems to conserved quantities. We define the conditionsgoverning such invariance. We wonder if the invariant quantities we exhibit can beinterpreted as sustainability. Using this method, we exhibit all the conservation lawsof the model, and we determine the conditions under which such laws exist. Weapply this method to canonic models with exhaustible resources, for whichsustainability issues are unavoidable. We develop two types of models consideringthe optimal intertemporal allocation of an exhaustible resource: a cake eatingeconomy as presented by Hotelling (1931) and production–consumption model asdeveloped in Dasgupta and Heal (1974) and Solow (1974).

We show that conservation laws exist under (very) restrictive conditions.Nevertheless, the invariant quantity exhibited in all models can be interpreted assome net production of the economy. We also show that if the optimal decisions are

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such that the net investment is nil (a general formulation of the Hartwick investmentrule), the utility is constant through time. The Hartwick investment rule (Hartwick,1977; Dixit et al., 1980) is thus a characteristic of constant utility paths. Note thatSato and Kim (2002) study the links between the Hartwick rule and economicconservation laws. Our result is complementary to theirs as it is more straightfor-ward and considers a more general model (we obtain a general formulation of theHartwick result without specification of a particular conservation law).

The paper is organized as follows. The Noether theorem and the link betweeninvariance and conservation laws are presented in Section 2. It is applied to a ‘cake-eating’ economy where the only good is a non-renewable natural resource in Section3. In Section 4, the invariance of a quite general production–consumption economywith two stocks – a stock of ‘man-made’ capital and a stock of natural resource – isexamined. An example is provided in Section 5, consisting in the simple model ofSolow (1974) and Dasgupta and Heal (1974), with no technological change and nocapital depreciation. An Appendix provides the mathematical contents and proofs.

2. Invariance and conservation laws

This section describes the Noether theorem and the way we use it to cope with thesustainability issue. One can refer to Noether (1918), Olver (1993) and Sato (1999)for more details on the Noether theorem and its applications in physical andeconomical frameworks.

The Noether theorem exhibits the invariance and conservation laws of adynamical system or, equivalently, it determines endogenous quantities that remainconstant along optimal paths of a dynamic optimization problem. This frameworkhas been largely used in physics but less in economics. Samuelson (1970a, b) was thefirst economist to seek conservation laws in the von Neumann growth model inwhich he showed the constancy of the ratio production/national wealth withoutusing the Noether theorem. Sato (1999, Chapter 7) obtained the same result usingthe Noether theorem and described how this theorem can be used to determine theconservation laws of growth theory models without environment. In the presentpaper, we apply this framework to growth models with non-renewable naturalresources.4 As mentioned in the introduction, Sato and Kim (2002) examine the linksbetween conservation laws and Hartwick’s investment rule, but without referring tothe Noether theorem.5

4Aulin (1997, 1998) also used a group theory (transformation properties of Lie’s groups) in his analysis

of economic growth. Askenazy (2003) described the Noether theorem in an optimal control framework.

We use this theorem in its original framework: the formal calculus of variations. As noted by Askenazy,

the calculus of variations is no longer used in economics, having been replaced by optimal control or

dynamic programming. Nevertheless, the optimal control framework used by Askenazy is not appropriate

to our approach as it requires knowing the symmetry properties of the system, which is precisely what we

try to exhibit.5They consider a particular conservation law (constant Hamiltonian in time-autonomous problems)

along a constant consumption path to again find Hartwick’s result and another investment rule for a

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Let us describe the Noether theorem more formally. Consider a dynamic system inwhich the state xðtÞ 2 Rn evolves under the dynamics _x ¼ gðx; u; tÞ, given decisionsuðtÞ 2 Rp, optimizing the following program6

maxuð:Þ¼

Z b

a

Lðt;xðtÞ; _xðtÞÞdt. (1)

We seek invariant quantities along the optimal path ½x�ð:Þ; u�ð:Þ� defined by thisprogram. If something is kept constant, the criterion could be proposed for copingwith the sustainability issue.

The Noether theorem determines the general principle relating the symmetrygroups and conservation laws of the variational problem (1). If one knows thesymmetry properties of the problem (1), the theorem leads to all the conservationlaws of the system.

Let us now describe the way it works. Consider a 1-parameter group oftransformations

t ¼ fðt;x; eÞ,

xi ¼ ciðt; x; eÞ; ði ¼ 1; . . . ; nÞ.

This group is a symmetry group if it keeps the integral (1) invariant, i.e. if

Z b

a

Lðt; xðtÞ; _xðtÞÞdt ¼

Z b

a

Lðt; xðtÞ; _xðtÞÞdtþ oðeÞ.

Every symmetry property is associated with a conservation law of the system (andreciprocally).

In economical terms, the transformation f can be considered as a ‘subjective time’(Samuelson, 1976). Similarly, the transformations ci can represent ‘technical’ or‘taste’ changes. These transformations are locally defined by their infinitesimalgenerators: t and xi, respectively, the first order coefficients of the Taylor series of fand ci around e ¼ 0. If the group of transformations is a symmetry group, theLagrangian of the problem, its derivatives and the infinitesimal generators of thetransformations satisfy the Fundamental Invariance Identity given by the equation

qL

qttþ

Xn

i¼1

qL

qxixiþ

qL

q _xi

dxi

dt� _xi dt

dt

� �� þ L

dtdt¼ 0. (2)

According to the Noether theorem, if the system has symmetry properties, theFundamental Invariance Identity leads to the conservation law of the system that is

(footnote continued)

constant utility. We exhibit all conservation laws thanks to Noether’s theorem, and show that when

Hartwick’s investment rule holds the utility is constant.6a and b are not necessarily finite.

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given by the following expression, which is constant through time:

O � L�Xn

i¼1

_xi qL

q _xi

!tþ

Xn

i¼1

qL

q _xixi, (3)

which is the same as

O � �HtþXn

i¼1

qL

q _xixi,

where H is the Hamiltonian of the problem. qL=q _xi is the shadow price of _xi and Ocan be interpreted as a measure of the change in well-being for infinitesimal changesof the state variables.

We are thus able to determine all the quantities that are invariant along theoptimal path if we find all the symmetry groups of the problem, i.e. transformationsthat satisfy the Fundamental Invariance Identity.

In the present paper, we want to determine the conditions for the existence ofconservation laws in a general resource allocation model based on the maximizationof an objective function which is the weighted sum of intertemporal utilities. Wefocus precisely on an exhaustible resource for which the sustainability issue isunavoidable. For this purpose, we need to exhibit the transformations associatedwith an invariant along the optimal path, and leading to a conservation law of thesystem. If there are such transformations under not too restrictive conditions,7 thecriterion examined could be proposed to tackle the sustainability issue.8 First of all,we argue that these conditions must not depend on a particular form of the utilityfunction.

Requirement 1. The existence of a conservation law does not depend on the form of the

utility function U.

In our approach, the sustainable development concept does not depend on the wayagents express their preferences. For example, any linear transformation of theutility function should lead to the same result.

3. A simple model: a ‘cake-eating’ economy

‘Cake-eating’ economy is often used to describe sustainability criteria (Heal,1998). Although its relevance to a policy debate is seriously limited, this model isessential to linking our approach to the literature in the field, and to intuitivelyexplaining the method (we use it like a ‘primer’ to describe how the Noether theoremworks).

7According to us, sustainability must not require the functions characterizing the economy to satisfy

unrealistic conditions.8I.e., the criterion could be discussed with respect to the ‘thing’ it optimally preserves.

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In this representation of the economy, a social planner determines an optimalintertemporal resource allocation by maximizing the sum of weighted per capita

utility. In our first model, the utility depends on both consumption and the stock ofnatural non-renewable resources. We seek conservation laws for the system. If thereare such laws, some quantities remain constant along the optimal path defined by thecriterion.

We consider the following model:

maxcð:Þ

Z 10

ZðtÞUðcðtÞ; sðtÞÞdt

s:t: _s ¼ �GðtÞcðtÞ. ð4Þ

U is the utility function depending on consumption ct and resource stock st. ZðtÞ isthe discount factor; we do not make any a priori assumption on it. We definefunction DðtÞ ¼ 1=GðtÞ which is a technical progress representing an improvement inresource use.

We examine under which conditions there are 1-parameter transformations

t ¼ tþ tðt; sÞe and s ¼ sþ xðt; sÞe,

that satisfy the Fundamental Invariance Identity defined by Eq. (2). It appears thatthe result depends on the resource extraction, i.e. if it is preserved (_s ¼ 0) or not, asstated in the following proposition and proved in the Appendix.

Proposition 1. (1) If _s ¼ 0, the transformation tðtÞ ¼ 1=ZðtÞ leads to the conservation

law O � U .(2) If _sa0, and if DðtÞ ¼ 1=ZðtÞ, the transformation tðtÞ ¼ 1=ZðtÞ leads to the

conservation law O � U þ _sDU 01 ¼ U � cU 01.

The first assertion of Proposition 1 represents the case where it is optimal topreserve the whole resource stock with no consumption. The conservation lawcorresponds to a constant utility: consumption is nil and the resource stock isconstant. The described path is the same as the optimal path given by the ‘greengolden rule’ (Chichilnisky et al., 1995). Moreover, we exhibit a relation between the‘subjective time’ and the discount factor. To link specifically our result to the ‘greengolden rule’, we can argue the following: maximizing the utility at an infinite timeimplies a ‘virtual’ discount factor ZðtÞ ¼ 0 for all t except for t!1. In this specialcase, subjective time tðtÞ is infinite for all finite times t, which means that the onlytime that matters (i.e. the ‘subjective time’) for the planner considering any time t isthe infinite horizon.

The second assertion of Proposition 1 states that if the optimal path is such thatresource consumption is positive, there is a conservation law only if the technologicalchange function DðtÞ is equal to the inverse of the discount factor ZðtÞ.9 This point inturn means that if a constant positive discount rate is assumed, one must also assumean exponential technological improvement (at the same rate). The technological

9It is that kind of relationship we judge unrealistic. The technological progress is probably linked to time

preferences of the planner, but certainly not so directly.

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progress must compensate for resource depletion. The conservation law of thesystem corresponds to the sum of utility and the loss of stock at consumption price(the marginal utility of consumption U 01) which is the definition of stock depreciationin Repetto et al. (1989).

This result is intuitive and very restrictive. Nevertheless, this model does notencompass all the aspects of the sustainability issue. To go forward in the analysis,we will now consider a production–consumption model in which accumulation ofproduced capital can, in some way, substitute for the availability of naturalresources.

4. A production–consumption economy

We now consider a production–consumption economy, in which utility Uðct; stÞ ofthe representative agent depends on consumption ct of a stock kt of ‘man-made’ capital, and on the existence of a natural resource stock st. Such a modelconsiders the interactions between production and resource use. The problem thenbecomes

maxcð:Þ;rð:Þ

¼

Z 10

ZðtÞUðCt;StÞdt. (5)

Capital stock evolves under dynamic

_k ¼ f ðk; rÞDðtÞ � ct � nk, (6)

where f ðk; rÞ is the production function (depending on the capital and the resourceextraction rt), nk the capital depreciation term and DðtÞ the technological progresswhich depends only on time. Resource stock evolves under the dynamic

_s ¼ �r. (7)

We apply the Noether theorem to the system (5)–(7). As before, we seek 1-parametertransformations

t ¼ tþ tðt; k; sÞe,

k ¼ k þ xðt; k; sÞe,

s ¼ sþ mðt; k; sÞe,

that satisfy the Fundamental Invariance Identity (Eq. (2)). Following the samemethod as in the previous section (the mathematical details are given in theAppendix), it appears that the results depend on the nature of the productionfunction. If we require the result to be valid for any production function(Requirement 2), we get Proposition 2 that is proved in the Appendix.

Requirement 2. The existence of a conservation law does not depend on the form of the

production function f ðk; rÞ.

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Proposition 2. There is no conservation law under Requirement 2 in the production– -

consumption model unless there is no technical progress, no consumption and no

extraction.

Proposition 2 states that there is a conservation law for the system, for allproduction functions, only if there are no extraction, no production, notechnological progress and no consumption. In other words, if we require the resultto be independent of a particular technology configuration, there is a family ofcontinuous transformations that have symmetry properties only if there is notechnical progress (DðtÞ is constant) and if _s ¼ 0 and _k ¼ 0. The transformationconsidered is t ¼ 1=Z and the quantity that remains constant is O � U . This case isnot very interesting from a sustainable development point of view as there is noconsumption or resource extraction.

We thus release Requirement 2 and examine two classes of production functions:we first look at a technology that is linear with respect to both inputs, and then at theCobb–Douglas production function. The results are, respectively, given inPropositions 3 and 4 and proved in the Appendix.

Proposition 3. If f ðk; rÞ ¼ ak þ br, and if DðtÞ ¼ n=ððn� aÞent þ aÞ and

ZðtÞ ¼ 1=ð1þDðtÞða=n2ÞtÞ, the transformations tðtÞ ¼ 1=ZðtÞ and xðt; k; sÞ ¼ DðtÞð1�ðb=nÞsÞ � ð _Z=Z2Þk lead to the conservation law

O � U þU 01ð_k þ _sDbÞ �U 01xðt; k; sÞ. (8)

Proposition 3 states that there is a conservation law only for a restrictive class oftechnological progress functions and if discount factor ZðtÞ satisfies a restrictivecondition close to a hyperbolic form involving technical progress. The conservedquantity is the expression of Eq. (3) for this model, which is the sum of theHamiltonian plus the infinitesimal effects of the state variable transformations (onthe capital stock here).

Eq. (8) is the same as

O � U þU 01ð_k � xðt; k; sÞÞ þU 01 _sDb.

Under this form, we can interpret the conservation law as the sum of utility andstocks depletion (at the consumption price), with the transformed man-made capitalstock _k � xðt; k; sÞ

� �.10

Proposition 4. If f ðk; rÞ ¼ karb, and if ZðtÞ ¼ �nent and DðtÞ ¼ enða�bÞt, the

transformations tðtÞ ¼ �ð1=nÞe�nt and xðt; kÞ ¼ ke�nt lead to the conservation law

O � U þU 01½_k þ Z_s� nk�,

where Z ¼ benða�bÞtkarb�1.

Note that Z is the marginal productivity of the resource, which is its shadow price.The quantity that remains constant along the optimal path is then the sum of the

10Note that x can be interpreted as a variation of the capital stock, due to the transformation on this

state variable.

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utility plus the loss of stocks, the term in brackets still being the stock depreciation(Repetto et al., 1989). If the optimal controls are such that the investment in capital _kis equal to the depreciation of the stocks ðr:f 0r þ nkÞ, then the term in brackets is niland utility is constant. We here have the Hartwick rule in a general framework.

Moreover, we here have a generalization of Proposition 4 in Stiglitz (1974). Ifthere is a capital depreciation, an exponential technical progress with a constant rateis needed. This result is valid for all conservation laws, and not only for preservingper capita consumption or growth. Nevertheless, the required discount factor is quiteunusual11 and would mean that the social planner minimizes the intertemporal sumof utilities if there is a positive capital depreciation rate.

5. The Dasgupta–Heal–Solow model

As a particular case of the previous general model, we can study the modeldeveloped by Dasgupta and Heal (1974) and Solow (1974), defined by the followingdynamics:

_k ¼ f ðk; rÞ � c, ð9Þ

_s ¼ �r, ð10Þ

where capital and resource stocks are denoted, respectively, k and s, extraction r andconsumption c. We consider a Cobb–Douglas production function f ðk; rÞ ¼ karb. Inthis model, there is no capital depreciation nor technological progress. The utilitydepends only on consumption.

We now consider the optimization problem defined by

maxcð:Þ;rð:Þ

Z 10

ZðtÞUðctÞdt, (11)

where ZðtÞ is the discount factor and UðctÞ the instantaneous utility of any form.12

So, we ask whether there are transformations

t ¼ tþ tðt; k; sÞe,

k ¼ k þ xðt; k; sÞe,

s ¼ sþ mðt; k; sÞe,

that satisfy the Fundamental Invariance Identity. This simpler model leads to moreintuitive results that are summarized in the following proposition and proved in theAppendix.

11Note that a constant and positive discount rate, as it is often assumed, would come from a ‘negative’

capital depreciation rate.12As noted by Robert Cairns (private communication), this problem is different from Solow’s maximin

problem. It is also different from the standard discounted utilitarian problem as we examine invariance

and do not take a specific discount factor. Moreover, we do not compute the optimal path or insure that

there is one. We just define what the invariance properties of the optimal path of the program would be.

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Proposition 5. In the optimization problem defined by (9)–(11), with f ðk; rÞ ¼ karb, ifZðtÞ ¼ 1=ðv1tþ w1Þ, the transformations tðtÞ ¼ v1tþ w1, xðkÞ ¼ v1k, and mðsÞ ¼ v2s,where v1; v2;w1 are constant parameters which satisfy the relation v1 ¼ ð�b=ðaþ bÞÞv2,lead to the conservation law

O � U þU 0½ _k þ _sf 0r � Zðxþ f 0rmÞ�. (12)

The time transformation t can be a translation and/or a dilatation. The solution ofthe problem is then not dependent on the chosen time scale. Actually, modeling aneconomy based on discrete time actions (speaking of an instantaneous productionmay seem strange) in a continuous time framework requires that the model isinvariant under group transformations on the time variable (contractions anddilatations in particular).13 In our context, this means that the sustainabledevelopment concept does not depend on the chosen time unity.14 Moreover, theresult is invariant under time translation. It means that considering our problem nowor in the future does not change the definition of ‘sustainability’.

In this model, we also exhibit the link between the subjective time tðtÞ and thediscounting factor ZðtÞ. This relationship leads to a hyperbolic discounting of theform ZðtÞ ¼ 1=ðv1tþ w1Þ. A hyperbolic discounting is much closer to the empiricalstudies than an exponential one (see Frederick et al., 2002 for a survey on the timediscounting issue) and would imply that the social planner has a subjective time closeto the empirically observed time preferences.

The second term of Eq. (12) is the value of change of the stocks, in utility units,given the transformations. Note that here again, the conservation of the utility levelwould require the optimal controls to be such that the investment is equal to theresource rents (notwithstanding the infinitesimal effect of the transformations).Thus, the Hartwick investment rule falls out of a Noether theorem argument quiteclearly in our model: constant utility paths in optimal growth problems arecharacterized by Hartwick’s investment rule.15 We exhibit the relationship betweenthe Hartwick rule and the conservation laws of the system in a way complementaryto Sato and Kim (2002).

6. Conclusion

In this study, we consider that sustainability requires that something be conservedthrough time. Moreover, we argue that the ‘thing’ to be preserved cannot be defineda priori, and we seek invariant quantities that are endogenous to a representation ofthe economy. We examine the sustainability issue by considering the intertemporal

13It is a well-known condition in physics. The scale of time must be invariant under a group of

transformations.14It is why we do not choose a discrete time model.15According to Cairns and Long (2006, Proposition 4), the discount rate supporting a maximin program

over an infinite horizon cannot be hyperbolic. Combining their result to ours means that the solution of

our problem is not a maximin path.

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allocation of an exhaustible resource, over an infinite time, as described in Heal(1998). We examine whether the neo-classical framework can provide such aninvariant. For this purpose, we seek conservation laws of quite simple but standardresource allocation models, including reproducible capital or not. We use theNoether theorem to describe the conditions under which there is an invariant alongthe optimal path of the dynamic optimization problem.

It turns out that there are invariant quantities that could be interpreted from asustainability point of view only under restrictive conditions. First of all, ineach model, the discount factor must have a particular form, which differs accordingto the structure of the economy (capital depreciation or not, form of theproduction function). It would then be hard to choose the ‘right’ discount factor toapply in a real economy. Moreover, in the simplest ‘cake-eating’ economy, thetechnological progress must be exactly the inverse of the discount factor. In a modelincluding reproducible capital, an invariant exists under strong hypotheses involving thetechnological progress and the discount factor once again. More specifically, in theCobb–Douglas case with a capital depreciation term, the technological progress has tobe of the exponential form and the discount factor must be negative and increasingthrough time. The Solow model without technological progress requires a hyperbolicdiscount factor.

We criticized the a priori conception of sustainability that consists in defining whatshould be maintained (consumption for example). We argued that thisapproach was not sufficient because it was based on considerations of the type:‘the development is sustainable if this function, or this other one, is of the form...’.We showed that the studied models lead to invariance only when such restrictionsare imposed. Thus, after this first work, we can argue that the theoretical frameworkof the described models (neo-classical optimization criterion) cannot define invariantquantities, even a posteriori, without formulating strong hypotheses on technologicalprogress or time preference. We thus wonder if the neo-classical approach makes itpossible to characterize sustainability as the conservation of anything throughouttime.

Anyway, in all the models we studied, the quantity that is invariant is equal to thesum of the utility (from the consumption and the resource stock level) and a termthat can be interpreted as the depreciation of the stocks valuated at a pricecorresponding to the marginal utility of consumption. This quantity can beinterpreted as the true income of the economy (Weitzman, 2003). We conjecture thatstudying more complicated economies will not provide more information, and thatthe more complicated the economy is, the more restrictive the conditions for theexistence of a conservation law will be.

Notwithstanding the restrictive conditions for such an invariant (that arethe main results of the study), optimizing such a criterion would lead to theconservation of a particular form of the net production of the economy (evaluated atsome shadow value). Moreover, if the decisions are such that the net investment iszero (the investment in capital compensate for the decreasing resource use)the quantity that is invariant is the utility. We here have a special form of theHartwick rule.

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From a more general point of view, we think that the use of the invarianceconcept could be an interesting way to reach an abstract definition of sustainability.A possible extension of this work should be to examine what isconserved by a competitive economy characterized by decentralized decisions,instead of considering a welfare-maximizing planner. Such an approach willmake it possible to ask whether decentralized economies can conserve anything inthe long run.

Acknowledgments

We thank John Hartwick and Robert Cairns for their comments and advice onprevious drafts of the paper. We are also much indebted to the invaluable commentsfrom the referees, which stimulated a substantial revision of an earlier manuscript.Any remaining shortcomings are our responsibility.

Appendix A

A.1. Proof of Proposition 1

The Lagrangian associated to system (4) is

Lðt; s; _sÞ ¼ ZðtÞU ½�_s D; s�.

We seek transformations

t ¼ tþ tðt; sÞe and s ¼ sþ xðt; sÞ e,

that satisfy the following Fundamental Invariance Identity defined from Eq. (2)

ð _ZU � _s _DZU 01Þtþ ZU 02x�DZU 01dxdt� _s

dtdt

� �þ ZU

dtdt¼ 0. (A.1)

According to Requirement 1, this identity must be satisfied for any utility function.Thus, we obtain the system

_Ztþ Zdtdt¼ 0, ðA:2Þ

_s _DZtþDZdxdt� _s

dtdt

� �¼ 0, ðA:3Þ

Zx ¼ 0. ðA:4Þ

Eq. (A.4) implies x ¼ 0. So, the system becomes


_sZ _Dt�Ddtdt

� �¼ 0. ðA:6Þ

ARTICLE IN PRESS


From Eq. (A.5), we get t ¼ 1=Z and dt=dt ¼ � _Z=Z2. Eq. (A.6) leads to twodifferent cases:

Case 1: If _s ¼ 0, Eq. (A.6) is satisfied for any technical change function DðtÞ.Eq. (A.1) is satisfied by the transformation t ¼ 1=Z and the conservation lawassociated with the problem is O � U , which is stated in part (1) of Proposition 1.

Case 2: If _sa0, combining Eqs. (A.5) and (A.6), we get

_D

D¼ �

_Z

Z¼) D ¼

1

Z.

Using Eq. (3), the transformation t ¼ 1=Z leads to the conservation law

O � U þ _sDU 01 ¼ U � cU 01.

It is the proof of part (2) of Proposition 1.

A.2. Proofs of Propositions 2– 4 of Section 4

The Lagrangian associated with the system (5)–(7) is

Lðt; ðk; sÞ; ð _k; _sÞÞ ¼ ZðtÞU ½DðtÞf ðk;�_sÞ � _k � nk; s�.

We seek transformations t, x and m that satisfy the following Fundamental Invariance

Identity:

qL

qttþ

qL

qkxþ

qL

qsmþ

qL

q _k

dxdt� _k

dtdt

� �þ

qL

q_sdmdt� _s

dtdt

� �þ L

dtdt¼ 0.

By differentiating the Lagrangian we obtain

ð _ZU þ _DZfU 0cÞtþ mZU 0s þ ZUdtdt

� ZU 0c ðn�Df 0kÞxþdxdt� _k

dtdtþDf 0r

dmdt� _s

dtdt

� �� ¼ 0.

According to Requirement 1, this equality must be satisfied for any utility function.We then get the following system:

_Ztþ Zdtdt¼ 0;

_Df tþ ðDf 0k � nÞxþDf 0r _sdtdt�

dxdtþ _k

dtdt¼ 0;

mZ ¼ 0 ¼) m ¼ 0:

8>>>><>>>>:

(A.7)

As the production function depends on _s, we must specify our requirements.Under Requirement 2 the factors of f and of its derivatives must be zero, which

lead to

_Dt ¼ 0,

Dx ¼ 0,

D_sdtdt¼ 0.

ARTICLE IN PRESS


As Da0, we get x ¼ 0. The only way to have ta0 is to set _D ¼ 0 and _s ¼ 0 (if t isalso equal to zero, there are no transformations leading to an invariant). The secondequation of system (A.7) implies _k ¼ 0. In this case, the time transformation t ¼ 1=Z

leads to the conservation law O � U , which is Proposition 2.Without Requirement 2, we need to choose some particular production function.

Writing dt=dt ¼ qt=qtþ _k qt=qk þ _s qt=qs and dx=dt ¼ qx=qtþ _k qx=qk þ _s qx=qs,the system (A.7) becomes

_Ztþ Zqtqtþ _kZ

qtqkþ _sZ

qtqs¼ 0;

_Df tþ xðDf 0k � nÞ þDf 0r _sqtqtþ _k

qtqkþ _s

qtqs

� �

�qxqtþ _k

qxqkþ _s

qxqs� _k

qtqtþ _k

qtqkþ _s

qtqs

� �� ¼ 0:

8>>>>>>><>>>>>>>:

(A.8)

These equations must not depend on _k and _s. As the production function depends on_s, we must specify its form before annulling the factors of _k and _s.

A.2.1. Linear production function

If f ðk;�_sÞ ¼ ak þ bð�_sÞ, system (A.8) becomes

_Ztþ Zqtqtþ _k Z

qtqkþ _sZ

qtqs¼ 0;

ðak � b_sÞ _Dtþ xðDa� nÞ þDb_sqtqtþ _k

qtqkþ _s

qtqs

� �

�qxqtþ _k

qxqkþ _s

qxqs� _k

qtqtþ _k

qtqkþ _s

qtqs

� �� ¼ 0:

8>>>>>>><>>>>>>>:

As t and x do not depend on _k and _s, the following system holds true

_Ztþ Zqtqt¼ 0;

qtqk¼ 0;

qtqs¼ 0;

ak _Dtþ xðDa� nÞ �qxqt¼ 0;

�b _DtþDbqtqt�

qxqs¼ 0;

qtqt¼

qxqk:

8>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>:

ARTICLE IN PRESS


The first equation means that t ¼ 1=Z, and we obtain the three following equations:

qxqt¼ akt _Dþ xðDa� nÞ, ðA:9Þ

qxqs¼ bðD_t� _DtÞ, ðA:10Þ

qxqk¼ _t. ðA:11Þ

We deduce from Eqs. (A.10) and (A.11) that x is linear on k and s, and then can bewritten under the form

xðt; k; sÞ ¼ x0ðtÞ þ _tk þ bðD_t� _DtÞs. (A.12)

Thus,

qxqt¼ x00 þ €tk þ bðD€t� €DtÞs. (A.13)

Combining Eqs. (A.12) and (A.13) to Eq. (A.9), we get the equality

x00 þ €tk þ bðD€t� €DtÞs ¼ akt _Dþ ðx0ðtÞ þ _tk þ bðD_t� _DtÞsÞðDa� nÞ

which can be reorganized as follows:

x00 � ðDa� nÞx0ðtÞ ¼ ðat _D� €tþ ðDa� nÞ_tÞk

þ ðbðD_t� _DtÞðDa� nÞ � bðD€t� €DtÞÞs.

Taking the various cross derivatives of Eqs. (A.9)–(A.11), we get the followingsystem from the last equation:

x00 � ðDa� nÞx0ðtÞ ¼ 0, ðA:14Þ

at _D� €tþ ðDa� nÞ_t ¼ 0, ðA:15Þ

ðD_t� _DtÞðDa� nÞ � ðD€t� €DtÞ ¼ 0. ðA:16Þ

Eq. (A.16) is equivalent to

D½ðDa� nÞ_t� €t� � _DtðDa� nÞ þ €Dt ¼ 0.

From Eq. (A.15) we get that ðDa� nÞ_t� €t ¼ �at _D. Eq. (A.16) then becomes�atD _D� _DtðDa� nÞ þ €Dt ¼ 0, or

�2aD _Dtþ n _Dtþ €Dt ¼ 0. (A.17)

ARTICLE IN PRESS


For ta0, integrating Eq. (A.17) leads to �aD2 þ nDþ _D ¼ 0 (for a nil constant ofintegration16), which is the same as _D=D ¼ Da� n. Eq. (A.14) then leads tox0 ¼ DðtÞ.

We also get the form of the technical progress, D ¼ n=ðgent þ aÞ, where theconstant from the integration g ¼ n� a if we set Dð0Þ ¼ 1. Note that the marginalproductivity of capital is positive only if a4n. Thus g is positive.

Furthermore, from Eq. (A.16) we also get the relation _D=D ¼ ðD€t� €DtÞ=ðD_t� _DtÞ. By integrating, we get D ¼ s1ðD_t� _DtÞ where s1 is a constant. Thisrelation leads to the first order differential equation

_t�_D

Dt ¼

1

s1. (A.18)

Note that without technical progress ( _D ¼ 0), we would have _t ¼ 1=s1, which leadsto a transformation t ¼ ð1=s1Þtþ s4, and thus to a hyperbolic discount factor asZ ¼ 1=t. We deduce from Eq. (A.18) the form of t:

tðtÞ ¼ DðtÞ s2 þ1

s1

Z1

DðtÞdt

� �

withRð1=DðtÞÞdt ¼ ðn� aÞent þ ða=nÞtþ s3.

As we have tðtÞ ¼ 1=ZðtÞ, we must have the following discount factor:

ZðtÞ ¼1

DðtÞðs2 þ 1=s1ððn� aÞent þ ða=nÞtþ s3ÞÞ.

Taking the particular values s1 ¼ n, s2 ¼ 0 and s3 ¼ a, the discount factor takes asimpler form

ZðtÞ ¼1

1þDðtÞða=n2Þt.

So, under the two transformations tðtÞ ¼ 1þDðtÞða=n2Þt and xðt; k; sÞ ¼ D�

ð _Z=Z2Þk � ðb=nÞDs, the conservation law of the system is

O � U þ _kU 01 þ _sDbU 01 �U 01 D�_Z

Z2k �

b

nDs

� �,

which is Proposition 3.

16If we do not choose a nil constant, the result is more complicated. For example, the technological

progress must have the form D ¼ 1=aðn=2þffiffi�p

tanðtffiffi�pþ oÞÞ where � and o are constants from the

integration. This is quite an unusual form of technological progress.

ARTICLE IN PRESS


A.2.2. Cobb– Douglas production function

For a Cobb–Douglas production function, the system (A.8) becomes

_Ztþ Zdtdt¼ 0) t ¼

1

Z;

kað�_sÞb _Dtþ xðDaka�1

ð�_sÞb � nÞ �Dbkað�_sÞb

dtdt

�qxqtþ _k

qxqkþ _s

qxqs� _k

dtdt

� �¼ 0:

8>>>>>>><>>>>>>>:

By eliminating the factors of _k and _s in the second equation, we obtain the followingequations:

ka _Dtþ xDaka�1�Dbka _t ¼ 0) x ¼

k

ab_t�

_D

Dt

� �, ðA:19Þ

qxqt¼ �nx, ðA:20Þ

qxqk¼ _t, ðA:21Þ

qxqs¼ 0. ðA:22Þ

Eq. (A.21) means that x ¼ k_t. So, from Eq. (A.19) we get a_t ¼ b_t� ð _D=DÞt andthen _D=D ¼ _t=tðb� aÞ. Moreover, from Eq. (A.20) we must have €tk ¼ �n_tk. Ifna0, it requires _t ¼ e�nt and t ¼ ð�1=nÞe�nt. Then, D ¼ enða�bÞt and Z ¼ �nent.17

The discount rate _Z=Z is equal to the depreciation rate n.Consequently, there is a group of transformations that satisfies the Fundamental

Invariance Identity: t ¼ ð�1=nÞe�nt and x ¼ ke�nt.The conservation law of the system is then given by the quantity

O � U þU 01½_k þ Z_s� nk�,

where Z ¼ benða�bÞtKarb�1 is the marginal productivity of the resource.

A.3. Proof of Proposition 5

The Lagrangian associated to Solow’s model is

Lðt; ðk; sÞ; ð _k; _sÞÞ ¼ ZðtÞUðkað�_sÞb � _kÞ.

17Again, if there is no capital depreciation, the transformations are linear and lead to a hyperbolic

discount factor. See details in Solow’s model case.

ARTICLE IN PRESS


It must satisfy the following Fundamental Invariance Identity:

_ZUtþ ZU 0aka�1ð�_sÞbx� ZU 0

dxdt� _k

dtdt

� �

� ZU 0bkað�_sÞb�1

dmdt� _k

dtdt

� �þ ZU

dtdt¼ 0. ðA:23Þ

As we want our result to be true for any utility function, this equation must notdepend on U or its derivatives. We thus obtain the following system:


aka�1ð�_sÞbx�

dxdtþ _k

dtdt� bka

ð�_sÞb�1dmdt� _s

dtdt

� �¼ 0. ðA:25Þ

From Eq. (A.24), we deduce that t only depends on t. By developing the timederivatives of x and m in Eq. (A.25), we obtain

aka�1ð�_sÞbx�

qxqt� _k

qxqt� _s

qxqtþ _k

dtdt

� bkað�_sÞb�1

qmqtþ _k

qmqtþ _s

qmqt� _s

dtdt

� �¼ 0. ðA:26Þ

As the infinitesimal generators of the transformations do not depend on the statevariations, Eq. (A.26) must not depend on _k and _s; the following system must holdtrue:

aka�1xþ bka dtdtþ bka qm

qs¼ 0, ðA:27Þ

dtdt¼

qxqk

, ðA:28Þ

qxqt¼

qxqs¼

qmqt¼

qmqk¼ 0. ðA:29Þ

We thus know from Eq. (A.29) that x depends only on k, and m on s. We can writet ¼ v1tþ w1, x ¼ v1k þ w3 and m ¼ v2sþ w2, where v1; v2;w1;w2;w3 are constantparameters. We can also deduce from Eq. (A.27) that v1 and v2 must satisfy therelation v1 ¼ ð�b=ðaþ bÞÞv2, and that w3 ¼ 0 (we also set w2 ¼ 0). Using theseresults with Eq. (A.24), we deduce that ZðtÞ must satisfy ZðtÞ ¼ 1=ðv1tþ w1Þ.

We can now determine the conservation law for the system. The quantity Odefined by the following expression is constant along the optimal path:

O � U þU 0½ _k þ _sf 0r � Zðxþ f 0rmÞ�. (A.30)

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