تغییر ناپذیری در نظریه رشد و توسعه پایدار

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.

In this paper, we propose a novel approach to sustainability. We search for endogenous invariant quantities in optimization problems. We then ask whether such constant quantities can be interpreted as ‘the thing to conserve for sustainability’. For this purpose, we examine and interpret the conditions under which these quantities remain constant. Let us detail our research program. In the economic literature, the sustainable development issue is often tackled in the way: ‘Something must be kept constant, or at least not decreasing’, and the debate is about the ‘thing’ to be preserved.1Solow (1993, pp. 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. Sustainability criteria can be used to determine what is to be conserved for sustainability (Heal, 1998). The most commonly used criterion is the neo-classical discounted utility View the MathML sourcemax∫0∞e-δtU(t)dt. Turn MathJax on This criterion is criticized mainly because the discount factor is decreasing2 and the criterion 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) and Stavins 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 objective function (in the criterion to optimize) but as an added constraint to an economic criterion. 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 consider the sustainability issue, and especially intergenerational equity. We agree with this point of view. For us, this approach is not relevant because it defines sustainability as an a priori constraint, leading to the following steps: a criterion is chosen (it is often the discounted utility maximization). Then, the constraints representing sustainability (constant consumption, non-decreasing utility...) are defined a priori. Finally, optimal paths are characterized. These paths are then interpreted thanks to the a priori definition of sustainability, leading to recommendations of the form: ‘The economy follows a sustainable path if3 the utility function (the production function, the pollution abatement function) is of the form...’. We reject this approach because the results are limited: how restrictions on the form of the utility function can be justified? Does the social planner have to impose its form to the representative agent? We do not think that an ad hoc utility function 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 can what has to be conserved for future generations be chosen a priori? The limit of this approach is that sustainability is considered both as a technical definition and a moral injunction. A definition that characterizes a particular development path as technically feasible (for instance having a non-decreasing consumption, or the possibility to preserve a natural resource stock) does not imply any moral strength to follow it. With such an approach, it comes out that positive propositions concerning the danger of a particular economical/environmental path cannot be separated from the 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 quantities endogenous to the representation of the economy. Such invariants will give a significance to the ‘thing’ we can and perhaps we want to preserve. We thus wonder what it is possible to sustain in a production–consumption economy. We adopt a general approach to find criterions of the form View the MathML sourcemax∫0∞Z(t)U(t)dt Turn MathJax on that lead to such invariant quantities. It is the general formulation of a program that maximizes some intertemporal sum of utilities. Z(t)Z(t) is a weightening function of any form (not necessarily of the usual exponential form). To summarize, we examine if the optimization of an utilitarian criterion makes it possible to define invariant quantities that could be interpreted as sustainability. We do not define what has to be preserved, and thus we make no explicit assumptions about the definition of sustainability. We consider the conditions which allow an essential neo-classical aggregated economy to sustain something. We especially examine conditions on the discount factor and technical change. For this purpose, we use the Noether theorem (Noether, 1918) to determine if optimal control problems – with utilitarian criteria and an exhaustible resource – lead to some invariant quantities along optimal paths. This theorem relates symmetries in dynamic optimization problems to conserved quantities. We define the conditions governing such invariance. We wonder if the invariant quantities we exhibit can be interpreted as sustainability. Using this method, we exhibit all the conservation laws of the model, and we determine the conditions under which such laws exist. We apply this method to canonic models with exhaustible resources, for which sustainability issues are unavoidable. We develop two types of models considering the optimal intertemporal allocation of an exhaustible resource: a cake eating economy as presented by Hotelling (1931) and production–consumption model as developed 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 as some net production of the economy. We also show that if the optimal decisions are such that the net investment is nil (a general formulation of the Hartwick investment rule), the utility is constant through time. The Hartwick investment rule (Hartwick, 1977 and Dixit et al., 1980) is thus a characteristic of constant utility paths. Note that Sato and Kim (2002) study the links between the Hartwick rule and economic conservation laws. Our result is complementary to theirs as it is more straightforward and considers a more general model (we obtain a general formulation of the Hartwick result without specification of a particular conservation law). The paper is organized as follows. The Noether theorem and the link between invariance 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 Section 3. In Section 4, the invariance of a quite general production–consumption economy with two stocks – a stock of ‘man-made’ capital and a stock of natural resource – is examined. An example is provided in Section 5, consisting in the simple model of Solow (1974) and Dasgupta and Heal (1974), with no technological change and no capital depreciation. An Appendix provides the mathematical contents and proofs.

In this study, we consider that sustainability requires that something be conserved through time. Moreover, we argue that the ‘thing’ to be preserved cannot be defined a priori, and we seek invariant quantities that are endogenous to a representation of the economy. We examine the sustainability issue by considering the intertemporal allocation of an exhaustible resource, over an infinite time, as described in Heal (1998). We examine whether the neo-classical framework can provide such an invariant. For this purpose, we seek conservation laws of quite simple but standard resource allocation models, including reproducible capital or not. We use the Noether theorem to describe the conditions under which there is an invariant along the optimal path of the dynamic optimization problem. It turns out that there are invariant quantities that could be interpreted from a sustainability point of view only under restrictive conditions. First of all, in each model, the discount factor must have a particular form, which differs according to the structure of the economy (capital depreciation or not, form of the production function). It would then be hard to choose the ‘right’ discount factor to apply in a real economy. Moreover, in the simplest ‘cake-eating’ economy, the technological progress must be exactly the inverse of the discount factor. In a model including reproducible capital, an invariant exists under strong hypotheses involving the technological progress and the discount factor once again. More specifically, in the Cobb–Douglas case with a capital depreciation term, the technological progress has to be of the exponential form and the discount factor must be negative and increasing through time. The Solow model without technological progress requires a hyperbolic discount factor. We criticized the a priori conception of sustainability that consists in defining what should be maintained (consumption for example). We argued that this approach 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 restrictions are imposed. Thus, after this first work, we can argue that the theoretical framework of the described models (neo-classical optimization criterion) cannot define invariant quantities, even a posteriori, without formulating strong hypotheses on technological progress or time preference. We thus wonder if the neo-classical approach makes it possible to characterize sustainability as the conservation of anything throughout time. Anyway, in all the models we studied, the quantity that is invariant is equal to the sum of the utility (from the consumption and the resource stock level) and a term that can be interpreted as the depreciation of the stocks valuated at a price corresponding to the marginal utility of consumption. This quantity can be interpreted as the true income of the economy (Weitzman, 2003). We conjecture that studying more complicated economies will not provide more information, and that the more complicated the economy is, the more restrictive the conditions for the existence of a conservation law will be. Notwithstanding the restrictive conditions for such an invariant (that are the main results of the study), optimizing such a criterion would lead to the conservation of a particular form of the net production of the economy (evaluated at some shadow value). Moreover, if the decisions are such that the net investment is zero (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 the Hartwick rule. From a more general point of view, we think that the use of the invariance concept could be an interesting way to reach an abstract definition of sustainability. A possible extension of this work should be to examine what is conserved by a competitive economy characterized by decentralized decisions, instead of considering a welfare-maximizing planner. Such an approach will make it possible to ask whether decentralized economies can conserve anything in the long run.