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Market structure and the search for exhaustible resources
Economics Letters 2 (1979) 85-90 0 North-Holland Publishing Company
85
MARKET STRUCTURE AND THE SEARCH FOR EXHAUSTIBLE
RESOURCES
Marion B. STEWART Rutgers College, New Brunswick, NJ 08903,
USA
Received March 1979
A dynamic programming model of mineral exploration and extraction is developed in order to examine the effects of market structure on exhaustible-resource industries. It is found that firms with market power rationally spend less on mineral exploration than would otherwise identical competitive firms, since the discovery of additional reserves has a detrimental effect on future prices which will be ignored by competitive firms.
1. Introduction
Until very recently, all studies in the economics of exhaustible resources - from Hotelling’s (1931) classic analysis of the subject through modern refinements and extensions of Hotelling’s work r - have assumed that the quantity of a resource available for production is known with certainty. The well-known ‘Hotelling rule’ is generated by entrepreneurs’ maximization of the constrained objective function
L =C
(1 + r)-‘(Pi - C)
Qi+ X(K- z. Qi)3
i=O
(1)
where Qi is the quantity produced in period i, Piis the price, C is the unit extraction cost (taken to be constant in this simple example), r is the periodic discount rate, and K is the (known) quantity of the resource available for production. The competitive solution to (1) is
(P,- c)= (1
+
r)(P,_l - c) for all
n > 0,
(2)
the ‘net price’ (price minus marginal cost) increases over time at the firm’s discount rate. Eq. (2) also defines the socially optimal production strategy, suggesting that the production of exhaustible resources can in principle be decentralized. (‘In principle’ since the social and private optimums will converge exactly only if social and private discount rates are equal and if firms do not attempt to maximize profits over a time horizon shorter than the economy’s lifetime. Neither of these assumptions seems likely to be satisfied in practice.) ’ For recent examples, see Stewart (1977), Stiglitz (1976), and Weinstein and Zeckhauser (1975).
86
M.B. Stewart /Market
structure and exhaustible resources
The presumption that (1) is indeed the objective function used by naturalresource firms leads to a surprising conclusion about the effects of market power. A firm sufficiently large to affect market price will maximize (1) by setting
(MR, - c)=(l+
r)(MR,_l - ct), n >O.
A monopolist thus chooses a prodyction strategy which leads net marginal revenue to increase over time at the firm’s discount rate. As Stiglitz (1976) has pointed out, however, if extraction costs are small, and if demand elasticities are approximately constant over time, then eqs. (2) and (2a) will be virtually identical: in this case ‘monopoly power’ is no power at all! It is clear, however, that this rather startling result is due to the very structure of the objective function (1). By assumption, a firm has a fixed (and known) quantity of a resource available for production, regardless of market structure, so that the monopolist’s usual profit-enhancing strategy - that of producing less than a competitive industry in order to keep prices high - is ruled out by construction. The monopolist’s only alternative, to change the time-pattern of its production, will not raise profits unless demand elasticities change systematically over time; so the effect of monopoly may be quite limited. The generality of this conclusion is severely constrained, however, by a serious defect - presumably widely understood, if not widely discussed - in the objective function (1). In most natural-resource industries - oil is an obvious example - the quantity available for eventual production is not known. In many extractive industries, geological estimates of the amount of the resource ‘in place’ vary by an order of magnitude or more; and in most industries it is not sufficient to regard K as the ‘expected’ quantity available, since even the expected value of K is itself a function of a firm’s exploration efforts. It appears that in these cases a different approach to the exhaustible-resources problem is required, involving a more sophisticated treatment of the subject of resource scarcity. A model in which the resource stock is taken to be a function of a firm’s exploration effort has been constructed by the author (1978), and independently by Pindyck (1978). The two papers reach somewhat similar conclusions, but differ in approach: Pindyck uses a continuous-time model in which all parameters are known with certainty, while the author uses a discrete-time model in which future prices, and the outcome of a firm’s exploration effort, are assumed to be uncertain. This discretetime analysis is presented below, in abbreviated form. Copies of the original discussion paper are available from the author.
2. A model of mineral exploration Let R,,be the stock of an extractable resource which a firm currently owns (or on which it has mining rights) and is capable of producing -- at some cost - over the remainder of its lifetime. The firm’s objective is to maximize the expected present
M.B. Stewart /Market
87
structure and exhaustible resources
value of profits, Z=zo
(1 + r)-'Eni,
given the initial resource stock R o.If indeed there is no more of the resource to be found, then the firm’s objective function is simply eq. (l), with expected price EPi replacing the deterministic Pi;and the expected-profit-maximizing strategy is just an expected-value variant of (2) or (2a). Such a model represents few if any existing natural-resource industries, however. Any firm possessing the appropriate technology can increase its stock of a resource by engaging in an exploration program. The relationship between exploration expenditures and additions to the firm’s stock of a resource - its ‘reserves’ - is of course a probabilistic one, since mineral exploration remains an imperfect science. There is also typically some time lag between exploration expenditures and reserve additions. For simplicity, assume that one period elapses between a firm’s exploration expenditure and the resultant addition to the resource stock. Mineral exploration is not inexpensive, and a firm’s decision to search for additional reserves of a resource must be regarded as an expression of its willingness to sacrifice current profits in order to generate greater profits in the future. Since a firm in an extractive industry must therefore choose both a production quantity and an exploration budget in every period, its objective will be the maximization of the function
Z= z$o(1 + r)wiEni=Fo( 1 + r)-'(E(PiC)Qi- WJi),
(3)
where Xi is the firm’s exploration effort in period i, and Wi is the unit cost of that effort. The objective function is subject to a resource constraint, Ri+i
=Ri-Qitf(Xi,~)>O,
(4)
where Riand Ri+lare the firm’s resource stocks at the beginning of periods i and i + 1, respectively, f(Xi, V) is the amount of the mineral resource ‘discovered’ during period i, and u is a random variable reflecting the uncertainty associated with mineral exploration. The solution to (3) and (4) is a straightforward dynamic programming problem, and we shall assume that a maximal value of (3) exists and that both Qi and Xi are non-zero (for some i > 0) in the optimal solution. ’ The maximal value of Z is
2 The existence of a finite quantity of the resource insures the convergence of (3) and the existence of a solution. If Ro > 0, then some Qi must be positive if Pi > 0. If mineral exploration is too expensive, or if entrepreneurs believe that the total stock of the resource has been found, then Xi = 0; we are interested, however, in the (apparently common) case in which neither of these conditions holds, so that some Xi > 0.
88
MB. Stewart
/Market
structure
and exhaustible
resources
clearly a function of the initial reserve stock Ro, so we may conveniently maximal value of (3) as ’ Z* = I”,(Ro)
=Max{(R,-, - C)Qe - ~eXe + cr
(1 +r)-‘R(pi
=(Po- CJQ; - wo& +(l +r)-%‘,(R1),
write the
- c?Qi-~Jil
(5)
where QG and XG are the optimal production quantity and exploration effort, respectively, in period 0, and Vr (R 1) is the maximal profit stream beginning in period 1, given in the resource stock R 1. The optimal production level Q(; must satisfy the first-order condition aZ*/aQ;=MRo-C-(1
+r)-‘EV;(R1)=O,
(6)
where MRo E PO + Q. aP,/aQ,, is the marginal revenue associated with PO; and V;(Rr)-=aV,/aR,. Now Vi (RI) is the marginal worth of the resource stock in period 1, which is clearly the net marginal revenue generated in that period. An additional unit of reserves available at the beginning of period 1 can be sold in that period, or it can be held for future sale. Profit maximization requires that, at the margin, the additional profit generated by immediate sale equal the discounted value of future profits which could be obtained if the resource were held for future sale. Thus the marginal worth of reserve additions in period 1 is the net marginal revenue in period 1, regardless of whether the additional reserves are sold to generate that revenue, or held to generate future profits. 4 Substituting Vi (R 1) = MR r - C into (6), we thus obtain the necessary condition for optimal production, MRo-
+r)-‘E(MR1
C=(l
- C).
(7)
This is, once again, the familiar Hotelling rule: net marginal revenue must grow at the firm’s discount rate, a result which apparently does not depend upon the assumption of a fixed stock of a natural resource. We now turn to the firm’s exploration program. Differentiating (5) with respect to XG, we obtain the first-order condition az*/ax;
= -wo t (1 t I--~E[v;(R~)
Substituting
af(x&
u)/ax;;]
= 0.
(8)
Vi (R 1) = MR 1 - C, we obtain
(1 +r)-‘E[(MR1
- c)af(x;,u)/ax;]
=we
.
(9)
3 Levhari and Srinivasan (1969) have used a similar dynamic programming approach to solve an optimal capital accumulation problem quite different from the problem considered here. 4 Proof. Differentiate (5) with respect to Ru (noting that R 1, Q& and X; are all functions of R,-,), and substitute from (6) and (8) to obtain V$Ro) = MRo - C, a result which is unaffected by an arbitrary change in the period index.
M.B. Stewart /Market
89
structure and exhaustible resources
The firm treats mineral exploration as any other factor of production, equating its expected marginal revenue product [the left-hand side of (9)] to the marginal explor. ation cost. Consider now the effect of market structure on mineral exploration. A firm operating in a competitive market chooses an exploration budget which satisfies (1 +r)_‘E[(Pr
- cjaj-(&,u)/aX;,]
(94
=wo.
But Pr > MX r , and assuming there are diminishing returns to exploration effort, ’ a competitive industry must therefore spend more on mineral exploration, ceteris paribus . In natural-resource industries, then, the burden of monopoly appears to fall primarily on the exploration function. Firms with market power rationally spend less on mineral exploration than would otherwise identical competitive firms, since the discovery of additional reserves has a detrimental effect on future prices which will be ignored by competitive firms (who, of course, regard their own exploration efforts as too small to affect price). The reduced exploration effort leads to a smaller stock of the resource, with a resultant increase in the spread between price and extraction cost in all future periods. Since the competitive solution is socially optimal (provided private and social discount rates are equal), 6 monopoly power in mineral exploration imposes a welfare loss on society in every period of its existence.
References Hotelling, H., 193 1, The economics of exhaustible resources, Journal of Political Economy 39, 137-17s. Levhari, D. and T. Srinivasan, 1969, Optimal savings under uncertainty, Review of Economic Studies 36, 153-162. Pindyck, R.S., 1978, The optimal exploration and production of non-renewable resources, Journal of Political Economy 86,841-861.
5 The assumption of diminishing returns is necessary, in any case, if (9) is to satisfy the secondorder conditions for a maximum. 6 Form the social welfare function
Qc; (go(Qi)-Odj-woX~+(l+r)-lESI(RI),
SO@O)=~
(A.11
0
where Sr(Ri) is the maximum discounted sum of consumers’ surpluses beginning in period i, given a resource stock of Ri; go( . ) is the market demand curve in period 0; and Qz and X6 are the socially optimal production level and exploration effort, respectively. Differentiating (A.l) with respect to Q6 and X6 yields PO - C =
(1 + r)-‘E(P1-
C)
and
(1 + r)-‘E[(Pf
the competitive market solutions implied by (7) and (9).
-
0 v-(x0,mxoi = WO,
90
M.B. Stewart /Market
structure and exhaustible resources
Stewart, M.B., 1977, Monopoly and the intertemporal production of a durable extractable resource, Department of Economics discussion paper, Aug. (Rutgers College, New Brunswick, NJ). Stewart, M.B., 1978, Market structure and the search for exhaustible Iesources, Department of Economics discussion paper, July (Rutgers College, New Brunswick, NJ). Stiglitz, J.E., 1976, Monopoly and the rate of extraction of exhaustible resources, Ameriban Economic Review 66,655661. Weinstein, M.C. and R.J. Zeckhauser, 1975, The optimal consumption of depletable natural resources, Quarterly Journal of Economics 89, 371-392.
85
MARKET STRUCTURE AND THE SEARCH FOR EXHAUSTIBLE
RESOURCES
Marion B. STEWART Rutgers College, New Brunswick, NJ 08903,
USA
Received March 1979
A dynamic programming model of mineral exploration and extraction is developed in order to examine the effects of market structure on exhaustible-resource industries. It is found that firms with market power rationally spend less on mineral exploration than would otherwise identical competitive firms, since the discovery of additional reserves has a detrimental effect on future prices which will be ignored by competitive firms.
1. Introduction
Until very recently, all studies in the economics of exhaustible resources - from Hotelling’s (1931) classic analysis of the subject through modern refinements and extensions of Hotelling’s work r - have assumed that the quantity of a resource available for production is known with certainty. The well-known ‘Hotelling rule’ is generated by entrepreneurs’ maximization of the constrained objective function
L =C
(1 + r)-‘(Pi - C)
Qi+ X(K- z. Qi)3
i=O
(1)
where Qi is the quantity produced in period i, Piis the price, C is the unit extraction cost (taken to be constant in this simple example), r is the periodic discount rate, and K is the (known) quantity of the resource available for production. The competitive solution to (1) is
(P,- c)= (1
+
r)(P,_l - c) for all
n > 0,
(2)
the ‘net price’ (price minus marginal cost) increases over time at the firm’s discount rate. Eq. (2) also defines the socially optimal production strategy, suggesting that the production of exhaustible resources can in principle be decentralized. (‘In principle’ since the social and private optimums will converge exactly only if social and private discount rates are equal and if firms do not attempt to maximize profits over a time horizon shorter than the economy’s lifetime. Neither of these assumptions seems likely to be satisfied in practice.) ’ For recent examples, see Stewart (1977), Stiglitz (1976), and Weinstein and Zeckhauser (1975).
86
M.B. Stewart /Market
structure and exhaustible resources
The presumption that (1) is indeed the objective function used by naturalresource firms leads to a surprising conclusion about the effects of market power. A firm sufficiently large to affect market price will maximize (1) by setting
(MR, - c)=(l+
r)(MR,_l - ct), n >O.
A monopolist thus chooses a prodyction strategy which leads net marginal revenue to increase over time at the firm’s discount rate. As Stiglitz (1976) has pointed out, however, if extraction costs are small, and if demand elasticities are approximately constant over time, then eqs. (2) and (2a) will be virtually identical: in this case ‘monopoly power’ is no power at all! It is clear, however, that this rather startling result is due to the very structure of the objective function (1). By assumption, a firm has a fixed (and known) quantity of a resource available for production, regardless of market structure, so that the monopolist’s usual profit-enhancing strategy - that of producing less than a competitive industry in order to keep prices high - is ruled out by construction. The monopolist’s only alternative, to change the time-pattern of its production, will not raise profits unless demand elasticities change systematically over time; so the effect of monopoly may be quite limited. The generality of this conclusion is severely constrained, however, by a serious defect - presumably widely understood, if not widely discussed - in the objective function (1). In most natural-resource industries - oil is an obvious example - the quantity available for eventual production is not known. In many extractive industries, geological estimates of the amount of the resource ‘in place’ vary by an order of magnitude or more; and in most industries it is not sufficient to regard K as the ‘expected’ quantity available, since even the expected value of K is itself a function of a firm’s exploration efforts. It appears that in these cases a different approach to the exhaustible-resources problem is required, involving a more sophisticated treatment of the subject of resource scarcity. A model in which the resource stock is taken to be a function of a firm’s exploration effort has been constructed by the author (1978), and independently by Pindyck (1978). The two papers reach somewhat similar conclusions, but differ in approach: Pindyck uses a continuous-time model in which all parameters are known with certainty, while the author uses a discrete-time model in which future prices, and the outcome of a firm’s exploration effort, are assumed to be uncertain. This discretetime analysis is presented below, in abbreviated form. Copies of the original discussion paper are available from the author.
2. A model of mineral exploration Let R,,be the stock of an extractable resource which a firm currently owns (or on which it has mining rights) and is capable of producing -- at some cost - over the remainder of its lifetime. The firm’s objective is to maximize the expected present
M.B. Stewart /Market
87
structure and exhaustible resources
value of profits, Z=zo
(1 + r)-'Eni,
given the initial resource stock R o.If indeed there is no more of the resource to be found, then the firm’s objective function is simply eq. (l), with expected price EPi replacing the deterministic Pi;and the expected-profit-maximizing strategy is just an expected-value variant of (2) or (2a). Such a model represents few if any existing natural-resource industries, however. Any firm possessing the appropriate technology can increase its stock of a resource by engaging in an exploration program. The relationship between exploration expenditures and additions to the firm’s stock of a resource - its ‘reserves’ - is of course a probabilistic one, since mineral exploration remains an imperfect science. There is also typically some time lag between exploration expenditures and reserve additions. For simplicity, assume that one period elapses between a firm’s exploration expenditure and the resultant addition to the resource stock. Mineral exploration is not inexpensive, and a firm’s decision to search for additional reserves of a resource must be regarded as an expression of its willingness to sacrifice current profits in order to generate greater profits in the future. Since a firm in an extractive industry must therefore choose both a production quantity and an exploration budget in every period, its objective will be the maximization of the function
Z= z$o(1 + r)wiEni=Fo( 1 + r)-'(E(PiC)Qi- WJi),
(3)
where Xi is the firm’s exploration effort in period i, and Wi is the unit cost of that effort. The objective function is subject to a resource constraint, Ri+i
=Ri-Qitf(Xi,~)>O,
(4)
where Riand Ri+lare the firm’s resource stocks at the beginning of periods i and i + 1, respectively, f(Xi, V) is the amount of the mineral resource ‘discovered’ during period i, and u is a random variable reflecting the uncertainty associated with mineral exploration. The solution to (3) and (4) is a straightforward dynamic programming problem, and we shall assume that a maximal value of (3) exists and that both Qi and Xi are non-zero (for some i > 0) in the optimal solution. ’ The maximal value of Z is
2 The existence of a finite quantity of the resource insures the convergence of (3) and the existence of a solution. If Ro > 0, then some Qi must be positive if Pi > 0. If mineral exploration is too expensive, or if entrepreneurs believe that the total stock of the resource has been found, then Xi = 0; we are interested, however, in the (apparently common) case in which neither of these conditions holds, so that some Xi > 0.
88
MB. Stewart
/Market
structure
and exhaustible
resources
clearly a function of the initial reserve stock Ro, so we may conveniently maximal value of (3) as ’ Z* = I”,(Ro)
=Max{(R,-, - C)Qe - ~eXe + cr
(1 +r)-‘R(pi
=(Po- CJQ; - wo& +(l +r)-%‘,(R1),
write the
- c?Qi-~Jil
(5)
where QG and XG are the optimal production quantity and exploration effort, respectively, in period 0, and Vr (R 1) is the maximal profit stream beginning in period 1, given in the resource stock R 1. The optimal production level Q(; must satisfy the first-order condition aZ*/aQ;=MRo-C-(1
+r)-‘EV;(R1)=O,
(6)
where MRo E PO + Q. aP,/aQ,, is the marginal revenue associated with PO; and V;(Rr)-=aV,/aR,. Now Vi (RI) is the marginal worth of the resource stock in period 1, which is clearly the net marginal revenue generated in that period. An additional unit of reserves available at the beginning of period 1 can be sold in that period, or it can be held for future sale. Profit maximization requires that, at the margin, the additional profit generated by immediate sale equal the discounted value of future profits which could be obtained if the resource were held for future sale. Thus the marginal worth of reserve additions in period 1 is the net marginal revenue in period 1, regardless of whether the additional reserves are sold to generate that revenue, or held to generate future profits. 4 Substituting Vi (R 1) = MR r - C into (6), we thus obtain the necessary condition for optimal production, MRo-
+r)-‘E(MR1
C=(l
- C).
(7)
This is, once again, the familiar Hotelling rule: net marginal revenue must grow at the firm’s discount rate, a result which apparently does not depend upon the assumption of a fixed stock of a natural resource. We now turn to the firm’s exploration program. Differentiating (5) with respect to XG, we obtain the first-order condition az*/ax;
= -wo t (1 t I--~E[v;(R~)
Substituting
af(x&
u)/ax;;]
= 0.
(8)
Vi (R 1) = MR 1 - C, we obtain
(1 +r)-‘E[(MR1
- c)af(x;,u)/ax;]
=we
.
(9)
3 Levhari and Srinivasan (1969) have used a similar dynamic programming approach to solve an optimal capital accumulation problem quite different from the problem considered here. 4 Proof. Differentiate (5) with respect to Ru (noting that R 1, Q& and X; are all functions of R,-,), and substitute from (6) and (8) to obtain V$Ro) = MRo - C, a result which is unaffected by an arbitrary change in the period index.
M.B. Stewart /Market
89
structure and exhaustible resources
The firm treats mineral exploration as any other factor of production, equating its expected marginal revenue product [the left-hand side of (9)] to the marginal explor. ation cost. Consider now the effect of market structure on mineral exploration. A firm operating in a competitive market chooses an exploration budget which satisfies (1 +r)_‘E[(Pr
- cjaj-(&,u)/aX;,]
(94
=wo.
But Pr > MX r , and assuming there are diminishing returns to exploration effort, ’ a competitive industry must therefore spend more on mineral exploration, ceteris paribus . In natural-resource industries, then, the burden of monopoly appears to fall primarily on the exploration function. Firms with market power rationally spend less on mineral exploration than would otherwise identical competitive firms, since the discovery of additional reserves has a detrimental effect on future prices which will be ignored by competitive firms (who, of course, regard their own exploration efforts as too small to affect price). The reduced exploration effort leads to a smaller stock of the resource, with a resultant increase in the spread between price and extraction cost in all future periods. Since the competitive solution is socially optimal (provided private and social discount rates are equal), 6 monopoly power in mineral exploration imposes a welfare loss on society in every period of its existence.
References Hotelling, H., 193 1, The economics of exhaustible resources, Journal of Political Economy 39, 137-17s. Levhari, D. and T. Srinivasan, 1969, Optimal savings under uncertainty, Review of Economic Studies 36, 153-162. Pindyck, R.S., 1978, The optimal exploration and production of non-renewable resources, Journal of Political Economy 86,841-861.
5 The assumption of diminishing returns is necessary, in any case, if (9) is to satisfy the secondorder conditions for a maximum. 6 Form the social welfare function
Qc; (go(Qi)-Odj-woX~+(l+r)-lESI(RI),
SO@O)=~
(A.11
0
where Sr(Ri) is the maximum discounted sum of consumers’ surpluses beginning in period i, given a resource stock of Ri; go( . ) is the market demand curve in period 0; and Qz and X6 are the socially optimal production level and exploration effort, respectively. Differentiating (A.l) with respect to Q6 and X6 yields PO - C =
(1 + r)-‘E(P1-
C)
and
(1 + r)-‘E[(Pf
the competitive market solutions implied by (7) and (9).
-
0 v-(x0,mxoi = WO,
90
M.B. Stewart /Market
structure and exhaustible resources
Stewart, M.B., 1977, Monopoly and the intertemporal production of a durable extractable resource, Department of Economics discussion paper, Aug. (Rutgers College, New Brunswick, NJ). Stewart, M.B., 1978, Market structure and the search for exhaustible Iesources, Department of Economics discussion paper, July (Rutgers College, New Brunswick, NJ). Stiglitz, J.E., 1976, Monopoly and the rate of extraction of exhaustible resources, Ameriban Economic Review 66,655661. Weinstein, M.C. and R.J. Zeckhauser, 1975, The optimal consumption of depletable natural resources, Quarterly Journal of Economics 89, 371-392.