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A contribution to the theory of depletable resource scarcity and its measures.

A CONTRIBUTION TO THE THEORY OF DEPLETABLE RESOURCE SCARCITY AND ITS MEASURES

I. INTRODUCTION

The question of devising suitable measures of the scarcity of natural resources has received considerable attention in the past quarter-century, beginning with the classic study of Barnett and Morse [1963]. This literature includes a long series of articles, of which the collection edited by Smith [1979] is an excellent example, and which, for the moment, culminates with the illuminating paper by Hall and Hall [1984].

The present paper expands upon Hall and Hall's review of the received literature. Following this, a stylized model of exhaustible resource production builds upon some of the classical insights. The model suggests that it may be worthwhile to de-emphasize the use of unit cost and of rent as measures of scarcity, at least in a Ricardian context where firms are physically constrained to mine varying qualities of ore simultaneously. It also de-emphasizes the importance of the question of ultimate exhaustibility of the resource. Rather, it emphasizes the role of capital, which is sometimes neglected in extraction models, and thereby the role of the exhaustibility of the deposit in influencing the choices of an extracting firm.

II. CLASSICAL AND CLASSIC TREATMENTS

Malthus and Ricardo, as elaborated by Barnett and Morse, differed in their treatment of agricultural land. Malthus [1798] assumed a fixed quantity of land,(1) so that food production could be expanded to feed a growing society at the extensive margin until the land constraint was reached. Thereafter, more food could be obtained only at higher costs by exploiting the intensive margin. Ricardo [1817] was at once more pessimistic and more optimistic. The quality of land was non-homogeneous. The best land would be farmed first, so that diminishing returns would result from any expansion; costs at the extensive and intensive margins would rise, and would continue rising as population increased.(2) While Ricardo did emphasize the role of the intensive margin, he did not discuss an ultimate limit, even though he was obviously aware of one. (Ricardo lived on an island and had corresponded with Malthus.) Instead, he advocated removal of trade barriers as a means of avoiding the need to move further into the extensive (as well as intensive) margin.

Hall and Hall [1984, 365-6] discern four types of scarcity in the non-renewable resource literature, namely, Malthusian and Ricardian flow and stock scarcities. Their discussion is summarized in Table I. The important questions are (1) whether a limit on total quantity produced is envisaged; (2) whether the resource is of uniform quality or whether there are depletion effects which cause quality to decrease through time; and (3) whether marginal extraction costs rise at any time as a result of constraints inherent in the characteristics of the resource. The distinguishing feature of Malthusian scarcities turns on the existence of an ultimate limit to total production, which in turn reflects Malthus's emphasis on the ultimate fixity of the quantity of cultivable land. The distinguishing feature of the Ricardian scarcities is that there is no ultimate limit to land, rather than diminishing returns at the extensive margin.[3] With these definitions, Malthusian flow scarcity is the most general concept.

There are three columns in Table I, with binary entries in each column. Thus, a total of [2.sup.3=8] "types of scarcity" can be imagined using this classification scheme. Table II extends Table I to include all eight possibilities and then categories the entries. Most of the categories have interpretations and are treated in the literature. Even more importantly, it is clearer than in Table I that attaching the labels "Malthusian" or "Ricardian" is sometimes suggestive but that trying to stretch the Malthusian and Ricardian notions developed for self-renewing agricultural land to mining can lead to inaccuracies and possibly to confusion. Table II suggests, for example, that "Ricardian Stock Scarcity" be identified not with Ricardo but with Mill [1848]. In chapter XII of book I, Mill was explicit about the operation of this effect at individual mines, whereas in his chapter III Ricardo [1817] was offhand, the point not being relevant to Ricardo's underlying purposes.

Table : TABLE I Hall and Hall's of Scarcity
 Costs same Costs same
 Ultimate within Time Through
 Limit? Period? Time?
Malthusian Stock Yes Yes Yes


Scarcity (MSS)

Malthusian Flow Yes No No Scarcity (MFS)

Ricardian Flow No No No Scarcity (RFS)

Ricardian Stock No No No Scarcity (RSS)

Table : TABLE II Extension of Table I
 Quality: Quality:
Ultimate Same Within Same
Limit? Time Period? Through time Reference in
 Yes Yes Yes (MSS)
 Yes Yes No Levhari & Liviatan
 [1977]
 Yes No Yes
 Yes No No (MFS)
 No Yes Yes (non-scarcity: a
 manufacturing, not
 resource, problem)
 No Yes No Jevons [1865]
 No No Yes (RFS)
 No No No Mill [1848]
 Solow & Wan [1976]
 (RSS)


Yet another type of scarcity could be identified with Jevons [1865]. He synthesized Malthus's theory of population and Ricardo's of diminishing returns in order to bring out the implications for a geographic region (Britain) of deterioration of its resource base (coal) as depletion progressed.(4) Jevons's main concern was a necessary move to higher cost coal, rather than early resource exhaustion.

Ricardo's consideration of the mine in addition to agricultural land is nevertheless germane to the subject of the present paper. He presents a contradictory pair of passages, one of which argues that his rent analysis applies to mining and one of which argues that it does not.(5) Ricardo's confusion arises because he is not certain how to deal with the ultimate exhaustion of the individual deposit (as opposed to all sources of the mineral in question) and its user cost in the context of varying quality. To resolve the confusion one must take a disaggregated view of the resource industry. As such Ricardo is a forerunner of Livernois and Uhler [1987], who identify flaws in the aggregative approach.

All these considerations lead to an expanded set of categories and authors in Table III. This table is doubtless not definitive. But it is suggestive of the points to be brought out in the present paper, whose single-mine model is also summarized in Table III. [TABULAR DATA OMITTED]

One of the points to be noted is that none of the four classical authors treats quality within the mine (e.g., the cut-off grade decision) at a given point of time, nor the comparable issue of efficient rates of extraction from an oil reservoir, as discussed by, for example, Davidson [1963]. One may argue that Ricardo should receive the credit on this matter. His discussion of the differentiated quality of agricultural land can easily be extended to encompass non-renewable resources. Moreover, in the cut-off grade problem the user cost appears in a very Ricardian way. Cairns [1986b] finds that it depends on differential quality and at the cut-off grade price is equal to marginal extraction cost.

A second point relates to total exhaustibility. Ricardo did not discuss non-renewable resources in terms of ultimate exhaustibility. Barnett and Morse, taking a "Ricardian" view, do not confront it. The non-renewable resource literature, treats the question of whether there is an ultimate limit to the resource as quite important. This is despite the fact that, if the world can be expected to take a long time, say more than a half-century, to exhaust the resource in question, then the discounted rent due to exhaustibility is essentially zero.(6) The world has not yet ever run out of a resource it was using. The reason is that Jevonsian scarcity engenders substitution (an effect which Jevons himself discounted). The substitution of natural gas for oil is imminent. But the substitution does not always run from one non-renewable resource to another, nor of a renewable resource for a non-renewable one; recall the substitution of steam for wind in sea transportation in the nineteenth century. It is clear that exhaustibility need not be the dominant constraint.

Furthermore, the analysis of an ultimate limit, in conjunction with Jevonsian deterioration, which is clearly present, requires a non-credible condition. If the resource is available at ever-increasing cost as its quality decreases, then cost, and therefore price, must approach infinity as "exhaustion" is approached. This notion is clearly non-operational: demand curves have intercepts in practice. As Solow [1974, 3] notes, no resource is "essential" in the mathematical sense.

There is a limit to the deposit, however. Even if the deposit can be reopened after several years of non-operation, the foreseen shut-down and the limit to its size requires consideration of the capital input. This is the subject of the model of the next section.

III. MODEL OF A MINING FIRM

We develop a partial equilibrium model of a price-taking mining firm. Price is determined endogenously by demand conditions and by the entry of new mines at the moment that maximizes discounted profits.(7) The ore deposit is shaped like a cylinder, with grade, g(r), diminishing with distance r from the axis. Extraction and concentration costs per unit ore, [C.sub.t], rise as mining progresses through the cylinder, but at any given instant there are constant returns to scale.(8)

[MATHEMATICAL EXPRESSION OMITTED]

where [H.sub.t] is the depth at which mining is taking place;

[h.sub.t] [equal to] [H.sub.t] is the rate at which depth is increasing; and

C'([H.sub.[tau]) > 0.

Total metal (or metal-in-concentrate) production is [pi] mh where

[MATHEMATICAL EXPRESSION OMITTED]

There is no returning to mine low-grade ores (at less than the marginal or cut-off grade at any time) once mining has progressed to greater depths. The cylinder is of total depth [H.sub.o]; beyond that depth the grade at any radius is zero.

Extraction and concentrating capacity is installed at the outset of production, at price P per unit.[9] Thus, volume mined is constrained:

[r.sup.2] h [is less than or equal to] K.

The objective of firm i is to maximize discounted profits from start-up ([S.sub.i]) to termination ([T.sub.i]). There is no confusion from suppressing the subscripts i (indexing deposits) and t (time), and the factor [pi]. The firm seeks to maximize.

[MATHEMATICAL EXPRESSION OMITTED]

In a full sectoral equilibrium model, start-up and termination times are endogenous. But under the assumptions of the present model, any investment occurs at start-up (S). For the individual price-taking firm, once the start-up time is determined from the endogenous future price path, analysis proceeds as if that time were exogenous. The Hamiltonian is pmh - [r.sup.2] hc - uh + v(K - [r.sup.2] h). The maximum principle yields the following conditions.

First order conditions: pm - [r.sup.2] c(H) - u - v [r.sup.2] = 0, (1) (pg - c(H) - v)h [equal to] 0. (2)

Dynamic condition:

[MATHEMATICAL EXPRESSION OMITTED]

Transversality conditions:

[MATHEMATICAL EXPRESSION OMITTED]

[MATHEMATICAL EXPRESSION OMITTED]

[MATHEMATICAL EXPRESSION OMITTED]

Equation (5) is determined from the more general transversality condition in what Takayama [1974, 656-60] calls the optimal control problem of Bolza-Hestenes. Its economic interpretation is that the mining firm invests in capacity up to the point at which the price of the marginal unit is equal to the discounted value of shadow quasi-rents earned by that unit.

Elimination of the Lagrange multiplier (v) from (1) and (2) yields

u [equal to] p(m - [r.sup.2] g) [equal to] pm(G - g)/G, (7)

where G [equal to] [m/r.sup.2] is the average grade of ore mined. This, of course, exceeds the cut-off grade (g), and (G - g)/G has an interpretation as the differential rent share in output. Since pm is the value of metal output, the user cost (u) represents the value in money terms of differential rent. Thus, Ricardian differential rent is identified with depletion of the orebody in the form of moving through the cylinder towards the terminal depth ([H.sub.T]). There is a trade-off between exploiting the extensive margin (increasing h) and exploiting the intensive margin (increasing r).

Solving (3) for the user cost when [MATHEMATICAL EXPRESSION OMITTED] K gives

[MATHEMATICAL EXPRESSION OMITTED]

Without further information about c'(H) it is not possible to characterize the path of user cost, and hence of differential rent, through time, at even the mine level, much less the sectoral level. Indeed, current rents may easily rise or fall through time. Recall that the resource constraint is expressed at the mine, not sectoral, level; this constraint arises even if prices remain constant for long periods. It is possible, however, to re-express (3') as

[MATHEMATICAL EXPRESSION OMITTED]

Discounted rents fall through time. Also, if cost is a constant function of depth then discounted rents remain constant (current rents rise at the rate of interest) at an individual mine.

Differentiating (7) with respect to time and using the result in conjunction with (3) yields

[MATHEMATICAL EXPRESSION OMITTED]

This equation provides some further justification for the inadequately explained appearance of [MATHEMATICAL EXPRESSION OMITTED] alone in the long-term estimates of changes in metal prices performed by Heal and Barrow [1981, 99-102]. Thus, to the extent that one accepts their estimation, there is some empirical support for the present model. But the justification comes from a Hotelling-style perspective of perfect foresight and equilibrium in asset markets, rather than from their disequilibrium perspective.

Equation (8) also indicates that the progress of the grade margin with price is of central importance. Since the grade margin in the model determines the "unit costs" of metal (or of metal in concentrate of some determined grade), the contribution of unit cost as a measure of scarcity, as apart from price, is not well-defined.

In fact, equation (2) (when h is not zero, so that mining is taking place) implies that

[MATHEMATICAL EXPRESSION OMITTED]

At the cut-off grade, metal value in the ore is exactly equal to marginal extraction costs plus quasi-rents to capital. Thus, the fact that mining is a capital-using activity as well as an ore-using one is crucial. The divergence between price and marginal extraction cost is the user cost of capital, not the user cost of the resource. In this sense, the cut-off grade is chosen so that the marginal grade just returns the cost of producing it.[10] The role of the interest rate is more complicated than in simple resource models, reflecting the two types of capital in the model, which give rise to (3') and (5), and the explicit grade choice which gives rise to a dynamic of exploiting the orebody reflected in (8).

By (5) the firm makes it capital stock choice at start-up (S) according to (a) the characteristics of the deposit and (b) future conditions that will hold over the deposit's life. Once that choice is made, the decision process involves the choice of a cut-off grade at each instant. This is the way that the resource constraint and user costs become a part of rational capacity choice and output decisions at the mine level. But the operation of the resource constraint may be masked by the fact, indicated by (1), (2) and (3), that resource scarcity is a phenomenon involving Ricardian differential rents; the firm determines the cut-off grade such that at the margin there is no scarcity rent apparent. The role of (4) and (6) is to determine whether the costs of mining are so great compared to the price of output as not to justify continuing to mine when the deposit is still "open at depth".

This characterization of the decision-making process corresponds to mining company practice in the daily determination of cut-off grades and in engineering analyses using discounted cash flow techniques in the investment decision. It is also consistent with statements by mining company executives that they do not actually take resource depletion into account in their decision-making.[11] Finally our results, though more stylized, are consistent with the empirical findings of Lasserre [1985] on capacity choice by mining firms.

Therefore, rents should not be used as a measure of sectoral scarcity: rents rise even if there is no true sectoral scarcity. In addition, as the deposit is depleted, unit costs will be rising: shut-down of the mine (but not the sector) will occur when mining cost at the attained depth, c(H), reaches the revenue obtainable from the highest grade of ore, at zero radius, pg(0). Again, unit cost cannot be used independently of price to measure scarcity.

Of course, the market price of output will reflect the sectoral depletion possibilities, as discussed by Cairns and Lasserre [1986]. But if extraction and concentrating costs are not rising quickly, as Cairns [1981; 1986a] found for nickel for example, then the capital stock choice will overshadow depletion in sectoral price determination.

Even if exhaustibility at the sectoral level should be important, the cut-off grade decision will not change qualitatively. Also, if sub-marginal`] ore which is not mined is "lost forever" - e.g through flooding, refillings, etc. of abandoned mines - then aggregate models based on the notion of progressing through a "world stock" distributed according to some probability distribution are not accurate and can be misleading. Rather, as Livernois and Uhler [1987] demonstrate for the oil industry, a disaggregated version is more representative. The effect discussed by Jevons is, then, both more complicated than in many current models and - to the extent that some mineral is lost - more compelling.

IV. CONCLUSION

This paper attempts to dispel the ideas (1) that either unit cost or rent is an unambiguous measure of scarcity independent of price in a non-renewable resource market and (2) that capital can be neglected in resource models. It also reiterates the importance of inframarginal sources in the measure of scarcity, and in the determination of a capital stock. In particular, Ricardian rents have an intertemporal allocational significance. This paper also returns thinking to Ricardian insights by highlighting the intensive margin of production of an orebody. It recognizes the importance of what Barnett and Morse identify as the Ricardian, but more properly might be called the Jevonsian, margin of production. For the sector as a whole, these considerations seem to dominate the more Malthusian concerns of total exhaustion.

The model remains, however, potentially misleading in its simplicity. What about cartels? Cairns [1986b] shows that a monopolist reduces both the quantity mined at present (h) and the cumulative quantity mined (H), as found in other models, but also increases the marginal grade (g). This is a second type of resource waste that might make a conservationist think twice about the virtues of slowing consumption. The model does not deal with externality, an expressed concern of Hall and Hall. It does not treat exploration or uncertainty; these raise other types of problems for the measure of resource scarcity.

( 1.) Malthus did not claim that the land was of homogeneous quality. Still, the homogeneity assumption would be consistent with his model. Consider Barnett and Morse's [1963, 56] remarks: "Malthus failed to take serious account of variations in the quality of agricultural land, or to pay attention to natural resources other than land ... For his purposes, the differences in the quality of the land could be ignored. So, too, could the question of the availability of resources other than land." Their model of Malthus's system [ibid., 60-61] assumes constant quality of land. One also notes the emphasis on the neglect of "resour es other than land".

( 2.) This is, of course, a highly stylized parable. For a more complicated analysis with the potential for contrary results, see Taurand and Hung [1987].

( 3.) In a model admitting exploration, where the "resource" is discovered and undiscovered oil pools, Livernois and Uhler [1987] identify search for new pools (of lower quality, in general) as exploiting the extensive margin and applying other factors to increase the flow rate at producing pools as exploiting the intensive margin. This industry (without exploration) is the example used by Hall and Hall [1984, 365] of Malthusian flow scarcity.

( 4.) Collison-Black [1981]. In his Chapter IX, "Of the Natural Law of Social Growth", Jevons [(1865) 1906, 194-5] made the following observation: "Malthus argued ... that though our numbers tend to increase in uniform ratio we cannot expect the same to take place with the supply of food ... The whole question turns upon the application of these views to the consumption of coal ... The momentous repeal of the Corn Laws throws us from corn upon coal."

( 5.) The two passages, from Ricardo's [(1817), 1953, 68, 85] Chapters II, "On Rent," and III, "On the Rent of Mines," are as follows: " ... the compensation given for the mine or quarry, is paid for the value of the coal or stone which can be removed from them, and has no connection with the original and indestructible powers of the land." "Mines, as well as land, generally pay a rent ... The return for capital from the poorest mine paying no rent, would regulate the rent of all the other more productive mines."

( 6.) A numerical example is given by Kay and Mirrlees [1975, 161-2].

( 7.) See Cairns and Lasserre [1986] for a model of the sector with capital investment but with homogeneous ore within each mine.

( 8.) Any further processing costs become part of c(H). Treating them explicitly would complicate the model, not adding to but obscurring the ideas to be brought out.

( 9.) This is a mathematical simplification, but such a choice is optimal under certain conditions - for example, if price remains constant. See Campbell [1980] for a proof of this assertion in similar circumstances, and Crabbe [1982] for further discussion. Slade [1988] finds that the price path of many minerals can be represented by a martingale.

(10.) It is easily checked that, as might be expected from other models, when grade is a constant function of radius, there are two "wedges" between average revenue (pg) and marginal cost (c) of ore, viz. the user costs of capital and the resource.

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*There are numerous reprintings of these works. The date refers to the dateof original publication.

(*) Department of Economics, McGill University and Centre de Recherche et Developpement en Economique, Universite de Montreal. I thank Philippe Crabbe and Tom Kompas for comments. Two referees and the managing editor were unusually helpful in improving the exposition. Work on this paper was supported by FCAR, Government of Quebec.
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Date:Oct 1, 1990
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