On Gray's Rule and the Stylized Facts of Non-Renewable Resources
Cairns, Robert D., Journal of Economic Issues
The economics of non-renewable resources is composed of at least two branches. The better-known branch owes much to a famous paper by Harold Hotelling , which most economists consider to be the first modern article in the field. Hotelling's main concern was with the unfolding of equilibrium through time in the market for a resource. Price at any instant of time was viewed as responding to the total resource stock available to society. As Robert Solow  pointed out, this response was a consequence of the fact that the resource was an asset to society and would in equilibrium have the same rate of return (through capital gains) as other assets. Price would adjust in intertemporal equilibrium so that the divergence between price (or marginal revenue if the market had market power) and marginal cost of extraction would rise at the rate of interest through time. (This follows because, if extracted and sold, a unit of resource would yield the going rate of interest. Only if the value of the resource rose by a like amount would it be left in the ground.) This result has come to be known as Hotelling's r-per-cent rule. Frederick Peterson and Anthony Fisher  have set out clearly the factors that are important to the Hotelling theory. These include the level of aggregate consumption, the level and rate of increase of price as The author is Professor of Economics, Megill University, and a member of the Centre de Recherche et Developpement Economique, Universite de Montreal and of the Laboraloire d'Economie Quantitative d'Aix-Marseille. The author wishes thank Carl Calantone, Gdrard Gaudet, John Hartwick, Anne Mayhew, Nguyen van Quyen, and a referee for comments and the FCAR and SSHRC for financial support. determinative of aggregate consumption, and the evolution of aggregate extraction costs as an influence on price, among others.
What is less often observed is that there is another tradition that abstracts from market equilibrium and deals with the "micromicro" aspects of production from deposits of non-renewable resources. This approach explains the effects of taxation and other public policies, investment, the physical and economic characteristics of the deposit, the size of the firm, and so on, on the rate of production from the individual mine or well. This branch looks into the black box of details neglected in the Hotelling branch. Almost a generation before Hotelling wrote, Lewis C. Gray [1913; 1914] discussed the relation between the two branches. He attempted to integrate the theory of the mine and the theory of optimal usage, optimal pricing, and the role of the interest rate.(1) His use of micro-micro considerations from the theory of the mine as the basis for the study of optimality and of market equilibrium provides a fugue for the present paper. Gray concluded that a (price-taking) firm facing u-shaped costs would adjust its rate of production (its marginal cost) so that the divergence between anticipated price and marginal cost would rise at the rate of interest. This would occur whatever the anticipated equilibrium path of price, the benchmark being a constant price through the life of a mine. This result could justifiably be called Gray's r-per-cent rule.
Meanwhile, in the Hotelling branch of the theory, an impasse has developed. Empirical studies have had (at best) mixed success in reconciling observation and theoretical predictions. As a result, several modifications--one might call them epicycles-have been proposed in an attempt to force observations to be consistent with theory. The object of the economics of non-renewable resources has been to describe what one observes by a series of stylized facts and then to explain these stylized facts in a way that is both sophisticated and interesting in the context of neoclassical economics. The thesis of the present paper is that three of the stylized facts need to be rethought.
What needs recognition is first that non-renewable resources are heterogeneous; this is a fundamental aspect to the rate of their exploitation, rather than exhaustibility. …