Hartwick's rule and maximin paths when the exhaustible resource has an amenity value

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Abstract

This paper studies the maximin paths of the canonical Dasgupta–Heal–Solow model when the stock of natural capital is a direct argument of well-being, besides consumption. Hartwick's rule then appears as an efficient tool to characterize solutions in a variety of settings. We start with the case without technical progress. We obtain an explicit solution of the maximin problem in the case where production and utility are Cobb–Douglas. When the utility function is CES with a low elasticity of substitution between consumption and natural capital, we show that it is optimal to preserve forever a critical level of natural capital, determined endogeneously. We then study how technical progress affects the optimal maximin paths, in the Cobb–Douglas utility case. On the long run path of the economy capital, production and consumption grow at a common constant rate, while the resource stock decreases at a constant rate and is therefore completely depleted in the very long run. A higher amenity value of the resource stock leads to faster economic growth, but to a lower long run rate of depletion. We then develop a complete analysis of the dynamics of the maximin problem when the sole source of well-being is consumption, and provide a numerical resolution of the model with resource amenity. The economy consumes, produces and invests less in the short run if the resource has an amenity value than if it does not, whereas it is the contrary in the medium and long runs. However, and without surprise, the resource stock remains for ever higher with resource amenity than without.

Introduction

Sustainability requires, broadly speaking, to maintain possibilities of well-being for future generations. But there is no real consensus on a more precise definition, and sustainability can be interpreted in many different manners.

In order to gain insights on the sustainability issue, the literature has mainly proceeded by studying various versions of the canonical Dasgupta–Heal–Solow model [5], [11], before resorting to more general set-ups. This model features a very simple economy using, besides man-made capital, the services of an exhaustible natural capital to produce its consumption good. The objective is to determine the intertemporal paths of depletion of natural capital and of accumulation of man-made capital, and whether natural capital must be entirely exhausted or not.

A first central issue is the one of substitutability between man-made capital and the exhaustible resource in production. The sustainability question is relevant only if this substitutability is poor, that is to say if the resource is essential in production [5]. This is the case in the Cobb–Douglas case which we consider in this paper.

Natural capital, besides providing productive services, has an amenity value for consumers. Krautkraemer [8] is one of the rare articles to examine a model where an exhaustible resource has an amenity value. An interpretation is to consider this resource to be natural capital, as its depletion appears to some extent irreversible. We follow this direction.

A second central issue, then, is the substitutability between consumption and the stock of natural capital in well-being. Weak sustainability admits some substitutability between consumption and the satisfaction derived from access to natural capital. To the contrary, strong sustainability implicitly considers this substitutability to be nil, and natural capital as the sole source of well-being.

A last central issue is the choice of the criterion of intertemporal welfare allowing to implement sustainability in the optimal growth model. Many agree to say that the usual discounted utilitarian criterion does not fit, as the process of discounting favors present generations at the expense of future ones. Other criteria have been proposed: undiscounted utilitarian criterion, maximin, Chichilnisky's criterion. Each one has its drawbacks, and none of them is widely recognized as the good one.

We focus here on the maximin criterion. It produces paths sustainable in the sense that well-being is held constant along them. The adoption of a maximin criterion in an intertemporal setting has been initiated by Solow [11]. He studies the Dasgupta and Heal [5] model in the case where the social objective is to find an equitable growth path. Equity is taken in the sense that the consumption of the least well-off generation is maximized. Solow confesses to be “plus rawlsien que le Rawls”, as Rawls himself was very reluctant to use a maximin welfare criterion for intertemporal problems [10]. Surprisingly, whereas the Solow model where utility only depends on consumption has been thoroughly studied in the literature, it has not been the case for the same model with natural capital as a direct argument of utility. Yet it seems natural as far as sustainability is concerned to express somehow that natural capital is a source of well-being besides consumption, and that both are not necessarily very substitutable. The first objective of this paper is to fill this gap in the literature.

From a methodogical point of view, we use a simple method to solve intertemporal maximin problems, which is simply to consider them as the limit case of a zero intertemporal elasticity of substitution. Moreover, we make a systematic use of Hartwick's rule [7] which states that the optimal investment rule, in a maximin context, is to conserve the total (man-made plus natural) value of capital for future generations. This allows us to obtain simple solutions, which has not much been done in the literature.

We obtain an explicit solution of the maximin problem in the Dasgupta–Heal–Solow framework, with a Cobb–Douglas production function, when the utility function is also Cobb–Douglas. We compute the optimal sustainable utility level and the paths of consumption, capital accumulation and resource depletion. Cobb–Douglas utility means a fairly high substitutability in preferences between consumption and the stock of resource, in the sense that it allows to sustain a positive utility level even though the stock of resource tends to zero. We then turn to the CES utility case, when substitutability in preferences between consumption and natural capital is lower. It is then necessary to preserve forever a critical level of natural capital, which will be determined endogeneously.

Then we study how technical progress affects the optimal maximin paths, in the Cobb–Douglas utility case. Contrary to [2], we make the usual assumption of an exponential technical progress. We give a complete characterization of the long run path of the economy, when both consumption and the stock of resource are arguments of utility. We show that capital, production and consumption grow at a common constant rate, related in a specific way to the rate of technical progress, while the resource stock decreases at another constant rate and is completely depleted in the long run. However, the higher the amenity value of the resource stock, the lower the long run rate of depletion. On the other hand, the physical rate of growth is an increasing function of the amenity value. Faster growth does not mean faster resource depletion.

We then build on a result derived by Cairns and Long [4] to develop a complete analysis of the dynamics of the maximin problem when the sole source of well-being is consumption. Unfortunately, this method does not apply in the general case of resource amenity and we have to resort to a numerical resolution of the model. The simulations show that initially the economy consumes, produces and invests less if the resource has an amenity value than if it does not, which allows slower depletion of the resource stock. Then, as this stock decreases, the economy must increase its reliance on physical production. Capital accumulation, with the help of technological progress, allows the economy to maintain a high rate of growth, without depleting too quickly the resource stock. The resource stock remains for ever higher than in the case of no resource amenity.

Section snippets

How to solve intertemporal maximin problems

Several methods are available to solve intertemporal maximin programs.

The problem is to maximizemintutunder technical constraints describing capital accumulation and resource depletion. ut denotes the utility level at time t, which depends on various variables.

Léonard and Long [9] state the objective as maximizingW=0ρe-ρtudt=ufor an arbitrary discount factor ρ>0, and add the following condition:ututto the technical constraints.

Cairns and Long [4] depart from this approach by directly

Maximin paths when the resource stock has an amenity value

Whereas optimal growth models dealing with pollution always take into account the desutility generated by its stock, models dealing with exhaustible resources almost never introduce the stock not yet extracted as an argument of well-being. This omission may seem rather natural if exhaustible resources are defined in a narrow way as fossil fuels or mineral ores. Yet, extraction and use of fossil fuels contribute to global warming and therefore have a negative amenity value. Stollery [12] thus

Technical progress and maximin paths

We now examine how technical progress affects the depletion of the resource stock. Does it allow to maintain a positive stock of resource, even in the Cobb–Douglas utility case? Should the physical capital stock still increase without limits, or can it converge to a finite level or even to zero as Solow [11] guessed, writing that “society asymptotically consumes its stock of capital as it consumes its pool of resource, relying on technical progress to maintain net output and consumption”, but

Conclusion

Hartwick's rule appears as a simple and useful rule governing the equitable use of resources and the compensating saving and investment effort required to preserve the well-being of future generations. We have shown that it applies in a wide range of situations, when the resource stock has an amenity value for consumers as well as when technical progress affects intergenerational trade-offs.

The use of simple models, with Cobb–Douglas or CES utility, enabled us to provide explicit or

Acknowledgments

We thank R.D. Cairns, N.V. Long, C. Withagen, participants at Paris 1 and Toulouse seminars, and two referees for useful comments, and especially M. Fleurbaey and C. Le Van for the proof in Appendix A.

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