THE THEORY OF THE MINING FIRM AND THE OPTIMAL EXTRACTION OF A NON-RENEWABLE RESOURCE*

NGO VAN LONG

Australian National University

I It is well-known that the traditional static theory of the firm cannot be applied

without modification to the analysis of the dynamic problems of a mining firm. The concept of 'user cost' has been used in the literature' to represent the dis- counted 'future' marginal profit foregone by a marginal increase in extraction now. Although this concept is useful for an intuitive understanding of the issues involved, the actual user cost itself, like many other shadow prices in optimiza- tion models, in general comes out of the solution of the problem rather than helps to obtain the solution. This paper presents a diagrammatic approach to a simple optimal extraction problem without recourse to an arbitrary construction of a user cost curve. Although our results can be obtained quite simply and definitely using optimal control techniques,] the approach adopted here has the advantage of being accessible to a larger section of economists. Moreover for reasonably simple problems, diagrams help intuition.

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(a) The Model Consider a fixed stock of resource, R o , which is accessible to only one firm.

We assume continuous time. Let t = 0 denote the present moment and T denote that point in time when the firm will cease its extraction. T is a decision variable for the firm and is not exogenously determined.

Let E(r) denote the rate of extraction at time t . The (instantaneous) average cost of extraction, C(E), is assumed to be a function of the extraction rate alone. For simplicity we assume away the possibility of storage's0 that all output E ( f ) is sold. The instantaneous demand curve P(E) facing the firm is assumed to be the same over time. The instantaneous average profit curve is obtained by subtracting the instantaneous average cost curve C(E) from the instantaneous average revenue curve (i.e. demand curve) P(E). When P(E) is downward sloping we call the firm a monopolist and when P(E) is horizontal we call it a competitive firm. In either case, we assume that the instantaneous average profit curve, denoted by n(E)'E, increases with E initially, reaches a maximum at E' and then decreases. In order

I am grateful to Professor J. D. Pitchford for valuable suggestions and to Professor L. Roy Webb for ointing out the analogy between the Ramsey saving rule and the firm's extraction rule. In particular.

Pam greatly indebted to F. Milne for stimulating discussion on this diagrammatic approach.

See Long 131, where optimal control techniques are used to analyse extraction paths in a macre economic framework.

' See. for example, Scott I 51, Davidson [ 1 I , Khoury [ 21 and Watkins 171.

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to avoid the trivial case when extraction is uneconomic, we assume that n(E)’E is non-negative for some E. This is depicted in Figure 1.

From the familiar average-marginal relationship, the instantaneous marginal profit curve n ’ Q crosses the average profit curve from above at E* and declines to zero at E m , the point where the instantaneous profit is maximized.

The firm’s objective is to maximize its (properly discounted) intertemporal profit, and not its instantaneous profit. Since cost and demand conditions are assumed to be the same over time, it is obvious that if the firm’s rate of discount is zero, its optimal extraction rate is a constant over time until time T , for there is no reason for it to be otherwise. For simplicity we shall assume that the firm’s rate of discount is zero, and note that a positive discount rate only tilts the optimal path of extraction toward the present.

FIGURE 1

Marginal Prolit

I

k r .-

2

Extraction Rate

We shall consider two cases: (i) There is no constraint on T (the point of time when the firm will cease its extraction) and (ii) T is constrained not to exceed some Fwhich is exogenously determined.

(b) Case one: T is unconstrained Given our assumptions of continuous time and of stationary demand and cost

conditions it is easily seen that the resource will be exhausted in this case. In order to prove this, suppose the firm ceases its extraction at time T and leaves some residual stock of resource R f . Noting that the optimal rate of extraction is is a constant, the firm’s total extraction is:3

and hence its profit over time is:

’ Since t = 0 is the present time, T is also the time intervalduring which the firm operates.

1973 THEORY OF THE MINING FIRM 295

But from (11, T can be increased, keeping E constant and hence a larger profit over time is possible. Therefore it is optimal to exhaust the resource. In other words when E and Ta re optimally chosen we have:

E.T = R , (3)

Our problem is to determine E (and hence T).

and (3): For any constant rate of extraction E, the total profit over time is, from (2)

v = Ro .n(E) / E (4)

It follows immediately that the rate of extraction which maximizes profit over time is E* (that rate which maximizes instantaneous average profit). This result is intuitively obvious: since the resource stock is fixed it is optimal to make the most profit out of each unit of the resource good.

We summarize our results in the following theorem: Theorem I : Given our assumptions, it is optimal to exhaust the resource. The

optimal rate of extraction is a constant, E: I t is that rate which maximizes instantaneous average profit.

Corollary I : The optimal rate of extraction is less than the rate which maxi- mizes instantaneous profit (except when the maximum average profit is zero, in which case E* = Em ). See Figure 1.

Corollary 2: A monopolist mining firm will produce at a point on the declining portion of its average cost curve.

Proof: At the point where instantaneous average profit is maximized, the slope of the (instantaneous) average cost curve must equal the slope of the (instanta- neous) average revenue (i.e. demand) curve, which is negatively sloped by assumption.

Corollary 3: (Scott) A competitive mining firm (i.e. its demand curve is hori- zontal) will produce at the point where its instantaneous average cost is at its minimum.

This result is due to Scott 151, without proof. It is a trivial corollary of our theorem 1.

We have so far assumed that the instantaneous average profit curve has a maximum at some point E+ which is positive. When this is not true, for example when the instantaneous average profit always declines as E increases, there exists no optimal extraction rate:4 For any constant positive extraction rate,

' See Long 131 and Vousden 161 for similar results.

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a smaller extraction rate will increase the profit over time, but zero extraction is non-optimal. This non-existence of an optimal extraction path is not present when the firm’s rate of discount is positive, in which case it can be seen in- tuitively (and can be shown mathematically) that the firm will start at some extraction rate E‘ and reduces it toward zero as the resource declines. Even when the rate of discount is zero, this non-existence of an optimal rate is not present if one assumes further that the extraction rate must be not less than a certain positive rate E‘ (in which case of course E‘ is the optimal extraction rate).

(C) CQSe two: T < T If the firm’s decision variable T is constrained to be not greater than a certain

7’,its optimal extraction rate will in general be different from E*. We shall consider three subcases:s

(a) T 2 T* where T* is the optimal exhaustion time when T is unconstrained; (b) T < T * d T>T, , where Tm is the time it takes to exhaust the resource if

the constant extraction rate is Em (the rate which maximizes instantaneous profit), and

( c ) T < T* and T Q T,.

Case (a) is trivial. E*is the optimal extraction rate and T* is the exhaustion time as before, for the constraint on T is not ‘tight’.

Case (b) implies that at the constant extraction rate Em the resource will be exhausted before the time T. We shall show that it is optimal to exhaust the resource in this case: For any time path of extraction which does not exhaust the resource by the time T , it follows from the above observation that in some time interval the extraction rate is less than Em. By increasing the rate of extraction toward Em, more profit will be earned. This shows that the non-exhaustion policy is non-optional.

We now show that, for case (b), the optimal exhaustion time is T itself. In other words, the optimal extraction rate is 6, where 6.F = Ro . This can be seen by noting that any constant extraction rate less than ,!? will not exhaust the resource by the time T, and hence is not optimal. On the other hand, any constant extraction rate greater than ,!? will yield less intertemporal profit than that obtained by extract- ing at the rate E, for while accumulated output is the same, the former extraction mode yields less profit per unit.

For case (c) we shall show that instantaneous profit maximization is optimal. We first show that except when T = T, it is non-optimal to exhaust the resource. For resource exhaustion implies that the extraction rate exceeds Em for some time interval (as R,, = EmTm 2 E,T by definition), and total profit over time can be increased by reducing the extraction rate in this time interval to Em .This argument also establishes the non-optimality of any extraction rate exceeding Em. On the

’ I am indebted.to N. Vousden for suggesting this approach.

1973 THEORY OF THE MINING FIRM 2w

other hand any extraction rate below E,,, is non-optimal, for over-all profit can be increased by increasing the extraction rate to E,,, (this does not reduce the profit in any period). Thus for ‘sufficiently short’ planning horizon (i.e. T < T, ) , instan- taneous profit maximization is optimal.

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We have been able to prove some intuitive results concerning the optimal extraction programme for a mining firm. Rather restrictive assumptions were made so that the problem is manageable using diagrammatic techniques alone. Various extensions can be made at the cost of mathematical complexity. It is hoped that this paper serves as an intuitive approach to the vastly complicated problems of resource exploitation.

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4. A. T. Scott, “The Theory of the Mine under Conditions of Certainty”, in M. M. Gaffney(ed.1.

5. A. T.Scott. “Notes on User Cost”, &XWIOmiC~OUrMl, vol. d, 1953. 6. Neil Vousden. “Basic Theoretical Issues of Resource Depletion”, Journal o/ Economic ,Theory,

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