Mineral Depletion and the Rules of Resource Dynamics.

• Author: Cairns, Robert D.
• Position: Report

1. INTRODUCTION

   Be a decision continuous or discrete, optimal natural-resource management is often presented in terms of an "r-percent rule." In keeping with the centrality of the rate of interest in a dynamic problem, such rules are conditions for how much to extract from a non-renewable resource, for when producers should enter or exit a market, for when a forest should be harvested, etc.

   Traditionally, the problem has been posed in terms of choosing a rate of flow from, or intensity of use of, a stock. Hotelling's (1931) rule states that in equilibrium the flow of an irreversibly extracted resource from a fixed stock is arbitraged over time such that its net price (price net of marginal extraction cost) rises at the rate of interest.

   An exception to analysis of flows is Faustmann's (1968) solution to "the tree-cutting problem." The decision is exogenously "lumpy:" it refers to an irreversible action on the whole stock. Timing, as opposed to intensity, is central. Faustmann's formula states that at the instant of harvest the net value of an optimally managed forest, consisting of land and trees, rises at the rate of interest.

   Irreversibility is a feature of both analyses. In the latter it differentiates the application of variable inputs from the deferrable application of capital (Davis and Cairns 2015). As with Faustmann's problem, the theory of deferrable irreversible investment has emphasized timing criteria for lumpy decisions. (1) A decision maker has options to operate on (harvest, plant, invest in, divest) an asset at different times. In the case of uncertainty the time to exercise is stochastic; a change in use is initiated as soon as the value of a state variable reaches a specified level that is determined from the assumed properties of the uncertainty faced by the decision maker. Resource problems have been center stage in providing much of the motivation for and many of the early results in the study of the timing of irreversible investments and other choices, and have been further used to show that r-percent relationships extend to any optimal timing action (Cairns and Davis 2007, Davis and Cairns 2012).

   Morris Adelman's paper, "Mineral Depletion with Special Reference to Petroleum" (Adelman 1990), is replete with insights with respect to optimal asset management. A succinct summary of his work to about the biblical age of "three score years and ten" (compare Adelman 1993), "Mineral Depletion" illuminates the multiple margins of decision in the oil industry that are components of an intricate equilibrium in oil markets over time.

   Adelman put relatively little emphasis on the instantaneous optimization of flow over time from an individual reserve: "current inputs are unrelated to current outputs" (1962, p. 3). Nor did he give credence to Hotelling's notion of a fixed stock. His interest lay in the timing of exploitation of oil reserves of various qualities, brought into working inventory (the stock) from the "much larger amount in the ground" (1993, p. xiii) as needed and as made possible by advances in technology. Reserves were scarce not because they were fixed but because they were expensive to replace, their value reflecting finding and development costs that may rise or fall over time.

   The present paper aims to provide a unifying view of the dynamic choice criteria of natural-resource use and thereby to position Adelman's contributions among several canonical papers in resource economics, renewable as well as non-renewable. We stress that, under certainty or uncertainty, an optimal decision consists of satisfying a particular type of r-percent rule: At the point of taking any incremental irreversible action the total return from not taking the action (the sum of the returns due to all attributes of the resource that would be left intact if the action were not taken) is equal to the rate of interest. In Hotelling's rule for flows, the total return is the capital gain to the marginal unit of the resource not extracted. In Faustmann's formula for a stock, the total return is the growth of the value of an entire forest. Expressing decisions in terms of r-percent rules unveils the fundamental economic features of the resource and its exploitation in capital market equilibrium.

   Though he rejected the r-percent rule for net price (Adelman 1986a), Adelman was aware of how the force of interest influenced the timing of decisions. We argue below that r-percent timing rules provide a framework for the insights in "Mineral Depletion." Such rules apply to decisions over all assets, including non-renewable and renewable resources. "A unit of [oil] inventory is an asset" (Mineral Depletion, p. 2). A dissection of the stock effect in non-renewable resource economics indicates that the rules also apply to the timing of use of reserves of different quality rather than to regulating flows of product from a single, uniform pool as in the usual interpretation of Hotelling's rule. Analogous conditions hold for non-optimally exploited resources. They apply to discrete as well as continuous decisions. They interact on multiple dimensions of decisions concerning a given resource. Furthermore, they apply, with slightly more complication, under conditions of uncertainty.

2. STOCKS, FLOWS, AND RENT

   Adelman's writing is dense with ideas and insights, and displays an emerging coherence of view about the equilibrium of assets, usually without assistance from formal models. His work, while firmly grounded in industry conditions, has been considered to be outside the mainstream, for two reasons.

   1. A pillar of his approach is the rejection of what he calls a fixed stock in favor of continual reserve replenishment ("Mineral Depletion:" 1-2). More broadly, Adelman denies the relevance of physical exhaustibility, a looming event to which other analyses are usually referred. He admits to the possibility of economic exhaustion nonchalantly: "investment dries up, and the industry disappears" ("Mineral Depletion:" 1).

   2. His formal model is in a partial-equilibrium tradition going back to Gray (1914) and Scott (1967). Output at a reserve is limited by fundamental technological and geological properties of production, viz., initial development and natural decline of the reserve's productivity.

   * The critical decisions are the level and timing of irreversible investment.

   * The operator is precluded from shifting output among time periods to realize Hotelling's rule.

   * Simply maintaining output demands continual development to replenish reserves.

   * Mathematical non-convexities impede aggregation to a sectorial equilibrium.

   Consequently, Adelman's notions of price paths, order of entry and extraction, etc., though cogent and highly descriptive of realized paths, are qualitative. They remain not fully formulated, not formalized.

   In this and the next two sections we argue that Adelman's insights ought not to be perceived as being outside the mainstream. A model of a renewable resource by Pindyck (1984) provides a fugue for our interpretation of Adelman's contributions.

   Pindyck considers the problem of exploiting a naturally growing stock of a resource, such as a fishery, under competitive conditions with property rights. (2) He assumes that the fisher maximizes the net present value, at rate r, of the net benefits of the fishery by modulating the flow of harvest. Since it is easier to catch fish if they are more abundant, unit cost is modeled as a decreasing function c(x) of the stock Q, a so-called stock effect, with c'(Q)

   That the resource stock has value, even though it is not fixed, accords with Adelman's view of oil reserves. Adelman was never able to pinpoint what it was that created reserve value--he vaguely refers to capital market equilibria that equate finding costs and present values of extraction, the latter assumed to be greater than zero in an apparent violation of the zero profit condition. Alfred Marshall would have recognized it as the quasi-rent necessary to induce investment. In Pindyck's model there is no investment and hence no quasi-rent upon extraction; the stock is continually and freely augmented. In his model stock effects are the source of reserve value. Adelman also subscribes to the dependence of cost on stock ("Mineral Depletion:" 3), but not, as we will show, in the same way Pindyck does.

   Pindyck interprets the condition for optimal use of the resource stock in three ways. In keeping with the theme of our paper we highlight the one that is an r-percent rule:

   [1]/[p-c] [d(p-c)]/[dt] + f'(Q) + [(-c'(Q))q]/[p-c] = r. (1)

   On the left-hand side, the total rate of return to holding rather than harvesting the marginal unit of stock is a sum of three rates, (1) the rate of capital gain on the marginal unit of stock, (2) the change in the rate of growth of the stock attributable to the marginal unit of stock and (3) the normalized reduction in harvesting cost attributable to the marginal unit of stock. This total rate of return at the margin is equated to the required rate of return on investment, the interest rate r, which is a parameter determined by the firm's situation in the wider economy. Since the rate of gain on the rent from leaving the marginal unit intact is the same as the rate of gain from harvesting it and investing the rent at the interest rate, the fisher is indifferent between harvesting a unit of stock now or later. Such indifference is a property of the optimal rate of flow.

   Pindyck's interpretation of optimality conditions in the framework of capital theory was innovative at the time. Adelman, too, viewed optimality conditions in such a framework: in interpreting Hotelling, Adelman (1986a: p. 324) writes, "the discounted net return from extracting a mineral unit from a given deposit in any year must equal that in any other year, which in turn equals any return from a holding with equal risk." At issue is what is thought to be creating that positive discounted return: quasi-rent...