Mineral Depletion and the Rules of Resource Dynamics

Abstract Conditions of exploitation of natural resources under certainty and uncertainty in some canonical natural resource problems are unified as r-percent rules by which the sum of all sources of gain from refraining from an irreversible action is compared to the interest rate. Action is initiated once the gain from action equals the gain from inaction. Morris Adelman’s insights, succinctly presented his 1990 paper on mineral depletion, are highlighted as implicitly recognizing, and even being grounded by, these timing rules.


I. 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. 1A 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 andDavis 2007, Davis andCairns 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 naturalresource 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.

II. 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. 2He 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 of the stock Q, a so-called stock effect, with cЈ(Q)Ͻ0 and cЈ(Q) r ∞ as Q r 0. Total c( ⋅ ) cost, represented as a linear function of the rate of harvest q, is written c(Q)q.The stock grows at the natural rate f (Q) net of the harvest: dQ/dt = f (Q) -q.The contribution to value is the product of the net price and output, [p -c(Q)]q.Because of the stock effect the marginal unit of resource has value, which in a steady state is "the capitalized value of future increases in cost resulting from a 1-unit reduction in the resource stock" (Pindyck 1984, p. 292).
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 valuehe 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: 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, stock effects, or the fixity of the stock.Adelman emphasized the first two, Pindyck's model the second, and Hotelling's model the third.
Other aspects of Pindyck's paper are consistent with Adelman's views.Pindyck (p.291) points out that the dynamics of his model are formally those of an exhaustible resource with reserve replacement and extraction costs subject to a stock effect. 3The focus of the analysis of a fishery is usually placed on attaining a steady state in which growth in reserves exactly replaces production.Adelman focused on reserve replacement as a central theme of analysis of oil markets.He did not suggest such a steady state, but did advocate against the possibility of physical exhaustion via stock effects ("Mineral Depletion:" 3).The stock effect is a shift upward of cost and marginal cost as the stock diminishes.With a stock effect, marginal cost may in time increase to reach the intercept of the demand curve so that no more resource is demanded.The result is sometimes called economic exhaustion, to which Adelman subscribes.The sectorial models of non-renewable resources with a stock effect are special, limiting, cases of his vision.

III. THE STOCK EFFECT RE-EXAMINED
The stock effect is a central theme in Adelman's discussions of the incentives to find and develop additional sources of oil supply.In this section we show that stock effects, too, can reveal r-percent rules for investment timing once resource discreteness is acknowledged.

III.1 Flaws in the Theory of the Stock Effect
Because models of the stock effect provide tractable sectorial equilibria, they have become mainstays of theoretical and empirical research.Where additions to the resource stock are admitted, as in Pindyck's model and in Adelman's view of oil and gas ("Mineral Depletion:" 3), even though the resource is not fixed the marginal unit may have value because of the stock effect.
Hotelling's simple model represents extraction cost as mq for a constant .The stock m ≥ 0 effect modifies the cost function to C(q,Q), assumed to be differentiable with and ∂C/∂q ≥ 0 Usually the effect is justified by the anticipation that better reserves are extracted earlier ∂C/∂QϽ0.so that there is a strict progression toward lower quality over time.Quality is perceived as being related to grade, depth or any physical feature that may cause extraction cost to vary among reserves.Occasionally it is noted that the unit cost of oil may increase because of decreases in pressure as a reserve is depleted.
In the model, however, the resource is portrayed as a single, homogeneous stock and the focus is on the intensity of flows of output from that stock.The objective is to maximize the net present value of the stock by choice of the flow q beginning from the current date, t: The Hamiltonian of the problem is In the solution, the shadow value k(t) rises at a rate less than r.Despite its aim to differentiate units of the stock, Problem (2) short-circuits the modeling of the order of extraction.In reality reserves occur in geographically separate, heterogeneous groupings called plays."The whole world is not one great play . .." ("Mineral Depletion:" 4).Nor are plays of differing qualities developed sequentially or qualities uniform throughout deposits, an idea that Adelman calls a "half-truth" ("Mineral Depletion:" 3).The technology and the pattern of demand over time affect which levels of quality, from which deposits, are in production at any time.They are not exogenous.The progression should be presented as an endogenous property of equilibrium rather than the immanent property of the path assumed by writing down what may be called the fictive cost function C(q,Q) (compare Slade 1988).Problem (2) and its Hamiltonian (3) imply that a single, homogeneous stock of size Q(t) has shadow value k(t) at time t.Cost is a property of the mass of original stock that has not yet been extracted, Q.It is inconsistent to maintain that there are multiple grades of mineral (what is the grade of the putative marginal unit to which the shadow value applies?), or that production is moving to greater depths (at what depth is the putative marginal unit?).As in Pindyck's fishery, it is implicit in the constraint (or ) that the units of the reserve are not physically distinguishable.
Problem (2) also assumes a fixed initial stock Q(0).Some authors have extended the problem by postulating a stock effect that applies to exploration as well as extraction.Exploration increases reserves by a quantity dQ, and extraction cost C(q,Q) is held to decrease: C(q,Q + dQ)ϽC(q,Q).Merely finding out that an oil reserve is larger than previously thought cannot be held to increase pressure.Therefore, the analogy to a single reserve does not hold; newly discovered reserves have to be considered to be at different locations.If quality varies, the addition dQ is of a higher quality than the reserves currently being produced.But that implication is contrary to what the analysts wish to model, that newly discovered reserves are of lower quality.The inconsistency of an aggregate model involving both an aggregate stock effect and exploration is examined by Livernois and Uhler (1987).They find two margins of decision that are traded off, extraction at the intensive margin and reserve additions at the extensive margin.The model becomes richer in the way that Adelman stresses ("Mineral Depletion:" 3, 4), though additions via exploration are still modeled as a continuum where intensity of effort, rather than timing of effort, remains central.
Another expression of the stock effect is one studied originally by Hotelling himself and extended by Gordon (1967), Levhari and Liviatan (1977) and Solow and Wan (1977).The change in cumulative extraction X(t), rather than in remaining reserves Q(t), is considered to be responsible for a steady increase in cost, now written .Adelman (1993: p. xii) makes mention of C(q(t),X(t)) the stock effect presented in this form: "at any given time, the more [any individual deposit] is developed, all else being equal, the higher the cost of each successive tranche" due to "rising marginal cost within the deposit."("Mineral Depletion:" 3).In a fixed-stock model (without exploration) one can put Q(t) = Q(0) -X(t).The shadow value of X(t) can be denoted by ; its rate of change is less than r.Thus, these expressions of the stock effect are not formally distinguished.If there is exploration, however, in the present case any newly found reserve must have lower quality than the reserve currently in production.This implication is broadly consistent with observations of a tendency of new reserves to be of decreasing quality.Livernois and Uhler's critique of the aggregate models leads them to disaggregate the flow.They find that the equilibrium when marginal cost is an increasing function of q exhibits multiple qualities in production at any time.The simple progression of the stock effect is broken, as Adelman understood.
The main flaw of these models of the stock effect is their limitation to modeling the intensity of flows, either in terms of extracting at the intensive margin or additions at the extensive margin.A more consistent grounding for the "half truth" of the stock effect can be gained by assuming that there is a multiplicity of small plays, each with a level of a physical property that can be summarized in its marginal cost.

III.2 A Continuous Model
Livernois and Uhler (1987) allow the number of stocks, strictly speaking an integer, to become continuous.The properties of the different reserves vary and the perception is that these reserves, though continuous, remain primarily stocks as opposed to flows.We use a similar device in this section to study a modification of Hotelling's simple model by Herfindahl (1967).Different qualities of the resource, with two or more different but constant marginal costs, occur in separate reserves.This modification is usually considered to be distinct from the stock effect. 4Herfindahl finds, as equilibrium conditions, that (a) reserves are exploited in strict order, lower costs earlier; (b) when a deposit is in production its net price rises at the rate of interest, an r-percent rule in the same form as the one in Hotelling's simple model with a single reserve; and (c) that at the transition between exploitation at different marginal costs the price is continuous but kinked, so that the price path through time is scalloped.
During the phase of extraction from a given reserve, the net price of a higher-cost reserve rises at a rate greater than r; it is optimally held back.The putative net price of a lower-cost reserve rises at less than r.There is a disincentive to hold it (Cairns and Davis 2007).Adelman (1993: p. xiv) expresses the economic reasoning exactly: "in a high-cost deposit the margin [of price] above cost will rise at more than the discount rate. ..The asset should be held, not produced.Contrariwise, a low-cost deposit. ..should be produced quickly." Let the reserves be indexed in order of unit cost by hϾ0.A firm holding a reserve with index h has an option to extract at the constant unit cost c(h) at any time t.The optimal path is an equilibrium of the exercise of these options.To study the implications of a smooth stock effect, let a limit be taken as the initial stock in each reserve is decreased and the number of reserves is 5.A number of such sequences can be proposed.For example, suppose that at stage n there are 2n -1 reserves having marginal costs , and that the total reserve R is the sum of the reserves at these deposits.The next stage c(h),h = 1, . . .,2n -1 is defined by inserting new reserves of quality , such that the total reserve remains R, (c(h) + c(h + 1))/2,h = 1, . . .,2n -2 distributed among 2(n + 1)-1 deposits.This procedure preserves total reserves but not present values.
6. Consider adjustments to production from two reserves that leave total output, and hence price, unchanged at each time.If the decision maker chose to reduce production at the higher quality reserve and to make it up from the lower quality resource, and vice versa at a later date, the net price of the units of higher-quality reserve so moved would rise at a rate less than r.The net price of the units of lower-quality reserve so moved would rise at a rate greater than r.
increased and distributed over a greater number of qualities (marginal costs), while leaving the total stock the same. 5In this limit, the initial size of each reserve becomes infinitesimal and reserves become distributed continuously over qualities.A continuous increase of marginal cost results as extraction proceeds through the infinitesimal reserves.In the limiting model a fictive cost function can be constructed and compared with the costs of the various physically distinguished units of mineral.
Each unit of the reserve with cost index h (now continuous) has a shadow value l(h,t) = l(h,0)e rt on the optimal path.It is distinct from the shadow values k(t) or introduced above.In k(t) each model of the sequence, including the limiting one, the shadow values express the conditions for the optimal timing of the exploitation of each reserve.Let represent the point in time at which t h the reserve of quality h is extracted.The first-order condition for reserve h while it is being exploited (when output q(h,t h )Ͼ0) is h For any other deposit with index h + y, yϾ0 or yϽ0 (for which in equilibrium), Therefore, the opportunity cost of producing from a reserve of any other quality is not h + y, y ≠ 0 covered by the equilibrium price .It is optimal to produce from a reserve when its net price p( t The rate of growth on the left-hand side is greater than r if yϽ0 and less than r if yϾ0.Before a reserve comes into production, its net price rises at a rate that is greater than the rate of interest.
Net price rises at the rate of interest while it is being extracted and at less than the rate of interest once it is exhausted.This relationship holds in the limit, where the difference y can be interpreted as infinitesimal.
7. The formal equivalence of Pindyck's model to a stock effect has in common with Herfindahl's model that the effect of quality is expressed through a unit-cost function, .c( ⋅ ) In this recasting of the stock effect according to Herfindahl's insights about varying quality and the opportunity cost of delay, each infinitesimal unit is viewed as a separate reserve, a capital asset managed optimally in reference to the rate of interest.Output from multiple qualities is not an equilibrium because of the assumption (related to the technology of production rather than the resource itself) of constant marginal cost at each reserve.At any time the true shadow value t ≤ t h of a unit of quality h is l(h,t).Its owner extracts the infinitesimal reserve of quality h at the equilibrium time according to the r-percent timing rule (4).The shadow value l(h,t) rises at t h exactly the rate of interest from the initial time 0 to the point of extraction .This is the set of t h conditions for the timing of entry and the order of entry into the industry in equilibrium.
In Problem (2), applied to this sort of non-renewable resource, the fictive cost function for the aggregate over the reserves in the flow analysis gives rise to a fictive schedule of shadow values k(t).If corresponds with , that schedule can be interpreted as the envelope of the true shadow values from the stock analysis.For reserve h, l(h,t ) . Competition minimizes the price of mineral at each time in the context of the demands of k(t ) h other periods and of the mineralization of different qualities-including reserves that are to be exploited eventually in the future."The market serves as a sensing-selective mechanism, scanning all deposits to take the cheapest increment or tranche into production ("Mineral Depletion:" 3).
The perspective from the analysis of timing is that there is not a single asset called the reserve for which a shadow value grows at a modified rate less than r.Rather, each unit of k(t) resource is an asset and exploitation of it entails a decision.The criterion makes use of a shadow value that rises at exactly r and that is specific to each unit.An order of extraction is determined, not exogenously but as the equilibrium among the options; each unit is extracted at the appropriate strike time.In "Mineral Depletion" (p.7), Adelman explains a similar, option interpretation of the movement to lower-quality reserves over time.
If there is exploration, the analyst must keep track of the quality of any newly discovered reserve and introduce it explicitly into the extraction sequence.Livernois and Uhler's (1987) device to do it in a tractable way for sectorial analysis and estimation is to model the effect of cumulative discoveries, N, as an argument of the extraction-cost function, written with C(q,Q,N) . Thus, their model contains a stock effect of the second type (in exploration) as ∂C(q,Q,N)/∂NϾ0 well as of the first (in extraction).The device gets them good mileage, but Adelman would consider even it a short-cut: in one of his earliest manuscripts on the subject he writes, "The cost of a successful effort [a discovery] cannot be ascribed to that effort alone but rather to the combination of all efforts, and the cost of all efforts, with which it was associated.Or, what comes to the same thing, the cost of the individual successful or unsuccessful project does not exist; it is a joint cost with other efforts of the same or even other companies" (1962: pp.6-7).
It is also possible to specify in another model that , a constant, for all h.It is then c(h) r m apparent that even the simple Hotelling model is at its root a stock rather than a flow model, with an infinite number of infinitesimal, separate reserves or assets.The condition on net price is a proxy for the true, inter-temporal equilibrium for assets, even for such ephemeral assets.The options view also implies that there is a comparable ephemeral, immaterial investment or disinvestment in Pindyck's fishery. 7

III.3 Summation
The true r-percent rules are masked in the traditional analysis using or as C(q,Q) C(q,X) the cost function.(For a fixed stock, recall, the interpretations are the same.)If there are units of different grade or unit cost, Herfindahl's method applies in the limit.The stock effect is transformed by a limiting process in traditional analyses from being based on stocks and the timing of entry to being based on the intensity of flow.A shadow value like or truly arises only if each unit k(t) k(t) of the reserve is physically equivalent, as in Pindyck's fishery.Invested capital has two necessary attributes, level and duration.Reality consists of lumpy reserves that are different in many ways, that are accessed by lasting investments, and that do not form a single reserve or play.The "whole truth" of the stock effect is a matter of timing.As Adelman ("Mineral Depletion:" 3) points out, the tendency to increasing costs is a complicated aggregate of many influences, with production from previously opened reserves continuing as others are opened.Analyzing extraction decisions as smooth distracts from the nature of the industry's decisions.Gordon (1967: 279) remarks that "a comprehensive description of behavior cannot be developed unless we grossly oversimplify our analysis."In passing to a tractable sectorial model from which testable (if spurious) hypotheses can be derived, the traditional stock effect gives up four essential properties of extractive industry: • within-reserve variation of quality, • simultaneous exploitation of reserves, • discreteness of investment and intensity of investment, and • non-convexity of technology and resulting occasional disruption of the market.
The example illustrates Adelman's fundamental and repeated insight, that the revelation of the operative r-percent rule can entail careful economic thinking before mathematical formality is introduced.Adelman does not abstract from the central questions of lumpiness of reserves and of investment.His instinct is that economic dynamics consists of a correct representation of the timing of decisions.

IV. ASSET VALUES
Adelman repeatedly commented over his career that in-ground oil had a value reflecting the present value of extraction, which producers equated to instantaneous finding and development cost at the margin.Because he rejected the fixed stock assumption, Adelman also rejected the idea that oil reserve values would necessarily rise over time.In fact, he felt that demonstrations that unit reserve values did not rise over time supported his dispatch of the fixed stock assumption: in-ground oil reserve values in the US were "essentially flat [in nominal terms] between 1947 and 1973," and that at a riskless rate of two percent, "[values] should have risen by 43 percent real in 1955-73" ("Mineral Depletion:" 4-5).But if the value of the marginal unit of resource is the net present value of its future extraction, such present value analysis requires that the marginal unit's value rise at the discount rate should holding the asset not benefit the investor in any other way.Pindyck's (1984: 293) analysis of a non-fixed resource reveals this r-percent relation.In his discussion of the stochastic version of his model, the social value of the fishery, V(Q), is the present value of the expected flow of net benefits from the harvest: Copyright ᭧ 2015 by the IAEE.All rights reserved.
8. Another example is that current production may be higher that the competitive level due to high private discount rates, a case that Adelman lays out in "Mineral Depletion." (5) ∫t In the notation of the present paper, his equation ( 9) can be expressed as Division by Vdt yields that, at the optimum, the sum of the expected capital gain on the stock, EdV/(Vdt), and what may be termed the rate of dividend flow from exploiting the fishery, q(p -c)/V, is equal to the interest rate, r.Even though Pindyck calls relation ( 6) the fundamental equation of optimality, it does not depend on optimization of the asset, but only on market equilibrium.Suppose that the fishery is subject to a resource-allocation mechanism (RAM) as postulated by Dasgupta and Ma ¨ler (2000).The RAM may not maximize the present value.Even so, it produces a unique trajectory over time.For any such trajectory let V(Q) be written as in equation ( 5) but not be maximized.Differentiation with respect to time yields relation ( 6).Even for a non-optimally exploited reserve, an r-percent relation holds for total capital value: the expected return on holding the asset is equal to the sum of dividend and capital gain, the two categories of return in this problem. 8Even though Adelman stresses that oil is not optimally extracted, he recognizes that "[t]he many kinds of investment are all substitutes for each other" ("Mineral Depletion:" 2).While he was referring to investments in developing oil, the concept is broader, applying to holding oil in the ground and to holding shares of companies.Since the latter are expected to provide an adequate return, so must the former.
Relation ( 6) is a simple implication of discounting to obtain present value.In equilibrium, any asset returns r percent through time.Testing whether an asset rises at r percent is not a test of whether that asset is a fixed stock resource.What Adelman misses is that the U.S. reserves he tracks are producing, and that the second term on the left-hand side of ( 6) is at play.Value V adjusts over time so that the capital market equilibrium obtains, in the case he examines requiring a real depreciation of the in-ground asset.Optimization of production does not change the relation in (6); it only shifts upwards the path for V.
Theorem I.For a freely traded asset, the rates of gain from all sources sum to the interest rate.The equality holds at the margin and for present values, for optimal and non-optimal management, under certainty and uncertainty.

V. THE GAIN OR LOSS TO POSTPONEMENT
Adelman uses reasoning about equilibrium timing to argue that lower-cost reserves of oil are developed earlier and that higher-cost reserves should be postponed ("Mineral Depletion:" 8).This is again an application of r-percent rules, though Adelman did not point this out explicitly.His analysis is reminiscent of Faustmann's formula for determining the optimal harvest age α = α 9.A comparable relation holds even if the harvest is not taken at the optimal time (Cairns 2014).As noted earlier, α the equality follows from the definition of value in the capital market, a result of discounting rather than optimality.
10.Such as harvesting at the age that the maximum sustainable yield is attained.
of a uniform forest.Let the net revenue from harvesting the trees at age α be represented by R(α) and the value of the land by L(α).The forest may also provide amenities, a dividend yield, over the rotation period.Let the total value of the flow of amenities be represented by .Suppose φ(α) that the instantaneous rate of interest r(α) is not constant and let the discount factor be represented by .The value of the (optimized) forest at age is the discounted value of harvesting it at α ≤ α the optimal time: ∫α ∂α Like any asset value, the value must obey an r-percent rule (must return rate r): 9 V(α,α) The asset value is not equal to the net realizable or intrinsic value from the program of V(α,α) harvesting at the current instant and also harvesting at the current age in future, 10 which is R(α) + L(α), nor to the value of harvesting at the current instant and harvesting at in future α rotations, which is : rather, it relates to the present value of optimized decisions, not to R(α) + L(α) the net realizable value.At age α, the premium from the option to harvest at instead of at α is α , the present value of the forest at the current date less the net realizable V(α,α) -[R(α) + L(α)] value.It reflects the value of the option to time the investment, where the option premium arises even under certainty.Adelman explicitly made this subtle point in his analysis of optimal timing ("Mineral Depletion:" 8).
The optimal harvesting condition (Hartman 1976) is that

΄ ΅ ˆR(α) + L(α) dα
If the forest has attained an age that is such that its growth and amenities are insufficient to justify holding it as a standing forest, the forest is harvested immediately.
the present value is increasing.The condition is a relation of the rate of growth of the net benefits expressed with respect to net realizable value, The forester should not plan to harvest at values of but to let the stand continue to provide α ˜Ͻ α its current flow of benefits, , which are greater than current interest on the net This analysis of a forest is true of the r-percent timing rule in general.

Theorem II. An asset may have several realizable attributes whose values may change through time. (i) It is optimal to act when the sum of the rates of growth of the realizable attributes, related to current net realizable value, is equal to the interest rate. (ii) The decision maker should continue to hold the asset (not act) if the sum of the rates of growth exceeds the interest rate. (iii) The decision maker should act immediately if the sum of the rates of growth does not exceed the interest rate.
This timing rule is an arbitrage or cost-benefit condition.It holds for decisions concerning the irreversible exploitation of any natural resource, and indeed any asset, under conditions of certainty (Cairns and Davis 2007).For oil, reserves are developed once the growth of realized present value reaches a trigger point where further postponement results in lost present value.Higher-cost deposits reach this trigger later: Adelman notes that "a 'dog' of a project is always worth postponing, and has only option value" ("Mineral Depletion:" 8).
As in the model of the stock effect, this perception from the theory of investment entails a conceptual break from much of natural resource economics.The reason for delay is not uncertainty, even though uncertainty constitutes an important generalization of previous theory.The reason is that it focuses on the timing of the development of stocks, not decisions over subsequent flows, in dynamic equilibrium.His perspective on resources having been informed by static Industrial Organization theory, Adelman struggled with this distinction but clearly had it in mind.The essence of resource economics, then, is that the value of an asset obeys an r-percent rule through time and that "full" cost includes the values of options.

VI. OTHER MARGINS
Missing from the usual discussion of the stock effect, but forcefully brought out in "Mineral Depletion," are (a) the idea of multiple margins, (b) the centrality of irreversibility to the minerals industry, and (c) the fact that investment is lumpy and lasting.These features introduce non-convexity and thereby influence the form and the very existence of market equilibrium in ways that are difficult to formalize.(Compare Cairns 2008.)For example, the model of forestry above formalizes a discrete harvest decision, as opposed to the flow decisions that are central to other resource models.
In "Mineral Depletion" (p.3), Adelman stresses the role of lumpy investment in the development of a reserve: "The new reserves in old fields were no gift of nature, but the payoff to development investment.""Most money is spent in development."His discussion of liquid hydrocarbons complements that of hard-rock minerals in a series of papers in the Canadian Journal of Economics in the early 1980s that extended a path-breaking article by Campbell (1980).In harmony with Campbell's model, Adelman's model of the exploitation of a single, given reserve ("Mineral Depletion:" 5ff) features the level of discrete investment in development (which determines the rate of natural decline of the reserve, a) as the only decision.Since output from a reserve of size R is q = aR, there is no subsequent flow decision, no response of output to the current price or interest rate.The fact that output does not respond to price was another of Adelman's points, mentioned as early as 1962 and confirmed empirically by Cairns and Davis (2001).Rather, the paths of price and interest determine the investment at the reserve-its timing and intensity.
The equilibrium into which development of a single reserve fits entails another r-percent rule.Before the reserve is developed, it has a market value V that aggregates its many characteristics, including the possibility of developing it at the appropriate time at the appropriate intensity for that time.Since no dividend is being earned on the undeveloped prospect, its total value is an option value that rises at the rate of interest: Under certainty or uncertainty, optimal or suboptimal management, , as outlined in Theorem I (Cairns and Davis 2007).
This r-percent rule for total value is the condition by which "[t]he many kinds of investment are all substitutes for each other, and the marginal cost of reserves added by one method should approach equality with the marginal cost of others ("Mineral Depletion:" 2)." "In thousands of projects, there are comparisons between the market value and the cost of reserve additions in reservoirs exploited; known but unexploited; incompletely known; suspected; hoped for. ..and so on (p.3)."The operative margin is not the cost margin, however, but the temporal margin, through which lumpy investments at the "thousands of projects" are brought into equilibrium at projected, endogenous prices.
The initial margin is exploration.It, too, is optimally timed according to the r-percent rule for total value as outlined in Theorem II.As per Theorem I it follows that .Because there V/V = r are many spillovers in exploration expenditure, the (total) net present value V is in this case not social value, but the projection of net present value of any appropriable production or knowledge that can be attributed to the exploration.
The market's "scanning-selective mechanism" indeed scans all deposits-producing, on the shelf and undiscovered.Smith's (2012) study of the margins of exploration, development and enhanced production indicates how intricate the decisions are at a single oil reserve, even with an exogenous price (even for a price taker).

VII. UNCERTAIN RESOURCE EXPLOITATION
Morris Adelman was well aware that the world was uncertain, often referring to "unknowable" futures and the randomness of growth of supply.In "Mineral Depletion" he makes note of Paddock, Siegel and Smith's (1988) analysis of oil reserves as option values.He focuses not on the idea that values under uncertainty also include an option premium but on the idea that options analysis reinforces concepts of optimal timing.In situating Adelman's contributions to resource economics, it is, then, of interest to indicate how some canonical papers in real options theory fit into the basic arbitrage or cost-benefit condition, the "r-percent rule," especially as conditions under uncertainty are not usually expressed as functions of time.Pindyck's (1984) model illustrates how uncertainty affects the r-percent rule when the intensity of flow is at issue: for risk-free interest rate r, and the expected-value operator E, the optimal harvesting decision yields 11. Equation ( 7) takes a slightly different perspective from Pindyck's; he moves this final term to the right and considers the algebraic sum of this term and the interest rate r to be a risk-adjusted rate.Yet it cannot reveal risk preferences, as the term vanishes if volatility of the stock, σ, is a constant.
12. The point-input, point-output model is nested in the model of Section VI above, with ϕ(α) ≡ 0.
Save for the final term on the left-hand side, this result is the same as in the analysis under certainty (see equation (1) above). 11The final term is an adjustment for the skewness of the random stock that arises from sigma being a function of the level of stock.A comparison with the Hotelling model is instructive: Hotelling's rule (under certainty) corresponds to the first term in equation ( 7): The other terms are zero because there is no uncertainty and there is no stock effect.
A different type of analysis extends the optimal timing of irreversibly and costlessly harvesting a point-input, point-output forest to the case of uncertainty (Clarke and Reed 1989). 12The risk-free discount rate is a constant, r, and there are two risk-neutral "drifts," (i) in the logarithm of the price of wood, b, and (ii) in the logarithm of the forest size, which is a decreasing function of age, α.The logarithm of forest value W behaves according to g( ⋅ ) 2 The net realizable value at age α, denoted by Y(W,α), is equal to e W .By Ito's Lemma the expected rate of growth of net realizable value (which in this case is the harvest value, since this is a point-input, point-output problem) at age α is The expected rate of growth of net realizable value is initially .Waiting is optimal. 1 2 b + g(0) + σ Ͼ r 2 Since monotonically declines, the forest is optimally harvested at the solution to In equation ( 9), the capital gain from changes in price (b), the growth in capital (dividend) from increases in wood content (g(α)) and the influence of Jensen's inequality on capital gains 1 2 σ 2 sum to the interest rate.
Equation ( 9) is non-stochastic and monotone in the sense that the expected rate of growth of net realizable value is not a function of the state variable W. In this model, uncertainty is not responsible for waiting and the condition for harvesting carries over directly from that for postponement under certainty. 13 Other considerations in stopping problems under uncertainty are flows of benefits or costs ("dividends") and stochastic influences that may affect choices of timing due to the value of information that comes from waiting, the quasi-option flow (cf.Davis and Cairns 2012).For example, if the stochastic growth of trees is size-dependent as opposed to age-dependent (Reed and Clarke 1990), harvesting delay to take advantage of quasi-option flow comes into play.In this more general setting, let the risk-adjusted interest rate at age α be represented by r(α).Let the net dividend (amenity) flow or cost (e.g., trimmings) flow again be represented by ϕ, and let the expected growth in realized value from stopping be stochastic.Generalizing a result in Davis and Cairns (2012), at the optimum stopping point, The partial derivatives with respect to age, and , are typically considered to be depreciation if negative (appreciation if positive) due to worsening (improving) project economics with the pure passage of time (Dixit and Pindyck 1994: 205-207).In the case of a finitely lived option, also V α includes the decay in the value of the option as the time to expiry draws nearer.The first two terms are thus the total rates of capital gains or losses from waiting to invest, stemming from changes in Y and α.The third term is the quasi-option flow (of information) from waiting to invest.The fourth term is the net rate of dividend received, or cost paid, by the option holder while waiting to invest.Equation ( 10) shows that at an interior stopping point, even for finitely lived options, the growth in value from all sources from delaying action, inclusive of expected capital gains or losses on net realizable value and any flows that may include value of information from waiting, equals the rate of interest.In Faustmann's formula under certainty, the r-percent rule corresponding with equation (10) holds when Y = R + L, the sum of the revenues from the forest and the land value.Again, the flow of amenities ϕ may be non-zero, in keeping with Hartman's (1976) analysis.
The other two components of Theorem II can be extended to the case of uncertainty as well.That is to say, while the firm is waiting, the expected growth in realizable value is greater than the rate of interest.

VIII. ABANDONMENT
In the exploitation of a forest under certainty there are two options, to harvest and to plant, that alternate through time.The harvest decision, discussed above, is a call option on the value of the timber and the land.The planting decision is a put option on the same variables.
A put option is discussed by Clarke and Reed (1990) in another context.They study the optimal time to abandon a perpetually producing oil well with fixed operating costs and declining 14.Clarke and Reed (1990: p. 364) are missing the negative sign in front of d.The error does not carry through the rest of their paper.production.Adelman (1962) noted two effects of low oil prices, at first a reduction in investment in reserve replacement and a consequent natural downward drift in production due to pressure declines, and eventually the abandonment of production at existing wells.This discrete decision is infrequently analyzed but of course is the counterpart of economic exhaustion in a model of oil production, and is among the decisions for which a temporal r-percent rule holds.
In what is known in petroleum engineering as the economic limit, abandonment of a reserve developed at time s occurs at a time T when the net value of production, p(T)q(T), falls to the fixed cost of production.Some of the reserve is never extracted.In the model in "Mineral Depletion," a quantity R(T) = q(T)/a is never extracted.
In contrast to the calculation of an economic limit, which presumes deterministic prices and production, Adelman took a forward-looking view to abandonment under uncertainty, recognizing that "even if price were slightly less than operating costs, the well would continue to operate if there were any hope of a better price in the not-too-distant future; for shutting down and reopening cost something-indeed, they may in extreme cases cost the well, because of sand clogging or water encroachment " (1962: p. 8).His latter statement stresses the irreversibility of such actions, an assumption that has been the mainstay of real options analysis of such decisions.Clarke and Reed, for example, recognize that to close the well is to effectively abandon it due to irreversibility of the decision.
In Clarke and Reed's notation, the oil price P(t) and extraction rate Q(t) evolve as geometric Brownian motions.Let and .Then p(t) = logP(t) q(t) = logQ(t) where b and d are risk-neutral drifts.Let denote the logarithm of revenue and let z(t) ≡ p(t) + q(t) .By Ito's lemma, z( t The goal is to determine the revenue level that induces optimal abandonment timing. 15.Because this is the solution of a dynamic problem that includes forward-looking optimization and adjustments for uncertainty, it is more general than the "economic limit" abandonment trigger .For example, only when σ 2 = 0, c 0 = 0 z ē = c and γ = 1 (no taxes) is (Clarke and Reed 1990: p. 368).
where .The current value of the well is and the Hamilton-Jacobi-Bellman (HJB) equation is According to Theorem II (equation 10), at the optimum At the optimal stopping point the realized value of further postponing abandonment is expected to rise, adjusted by a term reflecting quasi-option flow, at r percent.Solving (12) for the revenue level at which it is optimal to abandon yields as Clarke and Reed (1990) obtain by the usual tradition of solving the HJB equation given the boundary conditions for optimality. 15 Since the abandonment trigger will be reached with probability one over the infinite z ē life of the well, the shadow value of the oil in the reserve is zero.The scarce resource, with a positive shadow value, is the pressure of the oil (cf Cairns and Davis 1998).Pressure both allows the oil to reach the surface and, in conjunction with the development investment, limits the quantity of output at any time .In this model the effect of pressure is incorporated in the level of t ∈(s,T) production Q(t) and the decline rate d.Declining pressure, the suspension of reserve replacement or pressure augmentation, and possibly well abandonment is the way the market reacts to dips in demand (Adelman 1962(Adelman : pp. 8-9, 1986b)).

IX. AN ASSESSMENT OF ADELMAN'S INSIGHTS
As the canonical problems discussed in the present paper indicate, the approach in "Mineral Depletion" is consonant with a contemporary, mainstream, neoclassical view of the resource industry.Stress is placed on discrete actions such as investing in capacity, however, and not the subsequent flow of output.As befits Adelman's background in Industrial Organization, the r-percent rule of resource economics is isomorphic to the theory of investment by a firm under both certainty and uncertainty.Each decision gives rise to an r-percent rule.The actions may be continuous or discrete.
The opportunity costs and trade-offs are summarized, in an arbitrage or cost-benefit rule, in Theorem II.For flows, at the decision point the algebraic sum of capital gains and stock effects associated with leaving a unit for harvest a short time dt later, normalized by the net realizable value, is compared to the cost of capital (the cost of time, r, applied to the realizable value forgone).If the sum is greater than the cost of capital, the decision to reap the realizable value in a discrete case is delayed.In a continuous case the marginal unit is left unrealized.If the sum is less than the cost of capital, there is a corner solution in the discrete case: the decision maker strikes immediately.
For investment decisions, the dynamic path may involve lumpy investments and "abrupt change" ("Mineral Depletion:" n. 2).Despite the lumpiness, during a smooth transition actions are taken when capital gains from inaction, net of any amenity costs or benefit flows, fall from being greater than to being equal to the interest rate.Delay creates option premia, as with the harvesting of a forest.
There is no conceptual break when uncertainty is introduced: Bellman's equation (rearranged), or the adjoint condition from stochastic optimal control, is a general r-percent rule.Under uncertainty, the revelation of information over time can provide an additional source of return, which is part of the total that is equated to the interest rate.
These findings have been illustrated in some canonical resource-timing problems under uncertainty, which have been important precursors to the theory of real options posed in more general contexts.The r-percent rule provides insight to the economics of decisions through time.It has been a vital contribution of resource economics to economic theory.
Adelman gives the outline of and works within a sophisticated vision that strongly advances the economics of natural resources.In the corpus of his work, as summarized in "Mineral Depletion," he takes what may be called an "IO approach" to the problem of extraction.Analysis is rooted in the influences of geology and technology on a firm's irreversible actions.While he is aware of the obvious fact that the earth is finite, so that the content of oil in the earth or in any individual reserve is limited, his focus on irreversible investment at an individual reserve reveals how exhaustibility is not and would never become an operative constraint.Natural decline, coupled with a quasi-fixed production cost, assures that a substantial portion of oil is left behind when a reserve is abandoned.With changes in knowledge or technology, previously abandoned reserves can be re-opened, multiple times, with opening and closing decisions resembling more those of forest harvesting and planting than physically running out.Since the oil is not exhausted its "Hotelling rent" is nil-in a reserve and a fortiori in the aggregate.The scarce resource, pressure, allows the oil, which has value at the surface of the earth, to be extracted.Irreversible investments are made (a) in knowledge and in exploration to access the pressure, (b) in phased development to release it and (c) in enhancements to augment it.The flow decision is atrophied by output limitations.Adelman (1962: p. 13) asserts that "no sane petroleum operator plans otherwise than in terms of the whole period of time over which he will first invest and then recoup his investment in a field."The body of his work stresses stocks and irreversible actions on those stocks.Adelman's perception is emergent and not fully formulated, however.He ponders the addition of units of reserves from various sources, stressing the equalization of the costs at the margin, rather than the addition of lumps of reserves.He may not have appreciated fully that the equilibrium among the different types of investment involves an equalization of rates of growth of present values that supersedes the marginal-cost rule for flows.A rejection of a fixed stock of oil does not remove the capital market equilibrium that applies to oil inventories, which Adelman recognizes as assets.A main contribution of his formal model of investment is that it implies a non-convexity that may undermine the stability of competition or even of the industry if there is excessive entry.Yet he remains optimistic about competition rather than, for example, giving full consideration to the effects of disruptive investment and the effects of industry consolidation on it.(Neither does one see derivations of formal, dynamic, sectorial equilibria for Faustmannian forests.Moreover, oil reserves, subject to natural decline, may return toward equilibrium more easily than hard-rock mines, in which production limits do not decline regularly.)He struggles to interpret equilibrium as involving an equalization of marginal costs of replacement, while at other points he realizes that a replacement-cost margin is not well defined.Ambivalence is evident in the consideration of replacement through exploration and of investment in knowledge, intangible investments which he views as engaged in a "tug of war with diminishing returns" (depletion).
For all lumpy, irreversible investments the point of decision is determined by an equalization of the rate of growth of the gains from inaction, an r-percent rule for each temporal margin.Adelman does occasionally refer to real options, noting that a seminal paper by Paddock, Siegel and Smith (1988) provides "a more refined estimate of resource rent" ("Mineral Depletion:" 7).But he does not apply them to greatest advantage.He can be forgiven this lapse: the depth of the concept is obscured by the presentation of real options in economics as solely a feature of uncertainty rather than primarily of the passage of time.More fundamentally, Adelman is a pioneer, advancing new ideas, some of which were subsequently formalized.
Adelman discusses or hints at all of these ideas, pointing toward a more sophisticated, if less easily formalized, equilibrium in resource and other markets.His economics is profound but the mathematics has not caught up.Meanwhile, "Mineral Depletion" stands among the foundational articles of natural-resource economics.

r
) is a Brownian motion with risk-neutral driftd = db and σ = cov(p,q) pq variance .14At abandonment, let Ͼ0 denote the present value of per-c avoided net of the abandonment cost c 0 ; 1-γ a gross proceeds tax rate; and Ͼ0 the after-tax expected present value of revenues foregone, where r is γ + d -σ /2 the riskless interest rate.The net realizable value of abandonment at t is show that the value of the option to abandon when is of the z