EXHAUSTIBLE RESOURCE EXTRACTION UNDER DEMAND HETEROGENEITY
by
Ujjayant Chakravorty, Darrell Krulce and James Roumasset*
ABSTRACT
This paper develops a general model of exhaustible resource extraction when there are multiple independent demands and multiple resources and grades. Resources are characterized by constant unit extraction costs and conversion costs to each demand. We characterize patterns of resource use over time based on the concept of absolute advantage of resources in specific demands and across all demands. Conditions under which resources will be used at the beginning of the planning horizon for all uses and at the end of the planning horizon as an endogenous backstop are derived. We show that under demand heterogeneity progressive increases in the stock of a resource (for example, through unexpected, exogenous discoveries) may completely alter the sequence of extraction. For example, if oil stocks are limited, coal may be the backstop resource. However, if new discoveries of oil are made, the economy may shift completely to oil. If further additional discoveries of oil are made, we may revert to using coal in sectors where it has absolute advantage and preserve oil for later periods. These results are in sharp contrast to models with one demand that predict increased use of a resource with additional discoveries. The solution to a simple two-demand two-resource case is characterized.
JEL Classification: D9, Q3, Q4
This Version: February 2002
*Respectively, Department of Economics, Emory University; Xiox Corporation and Department of Economics, University of Hawaii at Manoa. Address for correspondence: .
EXHAUSTIBLE RESOURCE EXTRACTION UNDER DEMAND HETEROGENEITY
1. Introduction
The literature on the extraction of exhaustible resources (Hotelling, 1931; Dasgupta and Heal 1974; Pindyck, 1978) has almost exclusively dealt with the time path of resource extraction and resource prices under the assumption of a single homogenous demand for the resource. Beginning with Herfindahl (1967) effort has been made to study the issue of extracting multiple grades of a resource, focusing in particular on the tinme sequence of extracting different grades (Solow and Wan, 1976; Kemp and Long, 1980; Lewis, 1982) again under the assumption of a single demand for the many grades of a resource.
Empirically, however, it is common knowledge among industry observers that the energy sector of an economy is composed of distinct sub-sectors characterized by use, such as transportation, electricity, commercial and residential energy, etc. Although there may be resource substitution between these various end-uses, at least in the short run, it may be plausible to assume that the demand for each sub-sector can be represented independently. Empirical observation also suggests that typically there is more than one exhaustible resource (and grade) being extracted simultaneously for meeting the diverse energy requirements of an economy.
Chakravorty and Krulce (1994) have developed a two demand and two resource model in an infinite horizon framework where one resource has absolute advantage over the other in both demands. They have shown that under these specific conditions, it will always be the case that a more expensive resource will be used for a finite time interval even though the cheaper resource is not exhausted, violating the well known “least cost principle” of exhaustible resource economics. However, they did not proceed to develop the full implications of the multiple demand framework. This paper generalizes their framework in several directions, by considering an arbitrary number of resources and demands. Each end use is characterized by a downward sloping demand function and each resource has a constant extraction cost and a fixed use-specific conversion cost. No assumption is made regarding the absolute advantage of any resource over other resources as in Chakravorty and Krulce (1994). Solutions of an infinite horizon maximization problem yield equilibrium relationships for a given resource for a given end use in terms of the royalty and cost characteristics of the resource.
We start by developing general propositions that govern the extraction of resources in this general framework. In particular, we develop two distinct definitions of absolute advantage. Resources may have absolute advantage over other resources in a specific demand or in all demands. A cost-dependent concept of “relative efficiency” of a resource is developed that determines the ordering of royalties and the order of extraction of resources for any given use. The present paper develops conditions under which a resource may be exclusively used for all demands at the beginning of the planning horizon. Similarly conditions under which a resource may be exclusively used at the end of the planning horizon are described, which results in an “endogenous” backstop technology. The two resource two demand case is completely characterized under conditions in which one resource may have absolute advantage in all uses, and when each resource has absolute advantage in a specific use. It is shown that a weak absolute advantage in each use is likely to lead to a single resource being used exclusively at the beginning. Comparative dynamics results are obtained to show that an exogenous addition to reserves of a resource may
In a related piece of work, Gaudet, Moreaux and Salant (2001), henceforth referred to as GMS, have solved a somewhat analogous problem where solid wastes are transported from urban centers to spatially distributed landfills. In their model, landfill capacity is exhaustible, and they are differentiated by transportation costs from each city. There are some similarities between our problem and theirs, as well as important differences. The similarity is that transportation costs from city to landfills can be thought of as conversion costs of resources to demands. The key difference is that resources may be differentiated by class (oil, coal) or by grade (different grades of oil) while landfills are homogenous except for their location.
The focus of their paper is on developing the general solution in a spatial setting and applying it to the case of set up costs. In our paper, we focus on the general solution as well as develop conditions based on exogenous model parameters under which certain patterns of resource extraction may occur at the beginning and at the end of the planning horizon. In addition, we completely characterize the two-resource two-demand solution. These results are not part of the GMS paper. In general, transportation costs and end-use specific conversion costs as in our model have an important difference. Conversion costs to all demands may be equal for resources of the same class, and this allows us to make a distinction between resources of the same grade (oil of different costs) and class (oil and coal) and develop a taxonomy that distinguishes between resource class and grade. Thus a Herfindahl-induced ordering of resource grades can be developed for each class, as shown in this paper. In later work it may be useful to develop a general model with both end-use specific transportation and conversion costs.
In what follows, section 2 develops the general Hotelling theory with multiple demands. Section 3 provides the complete solution to the two demand two resource model. Section 4 concludes the paper by highlighting the usefulness of a multi-demand multi resource approach over the traditional single demand Hotelling framework.
2. A Model of Multiple Resources and Heterogenous Demands
Consider a finite set of resources R (such as oil, coal, natural gas, etc.) and a finite set of uses for these resources defined by the set U (such as electricity, heating, transportation, etc.). The available stock of resource I Є R is qi(t0)>0which can be extracted at a unit cost of ci ≥0. Demand for use j Є U is a strictly positive. Bounded, continuous, strictly decreasing function of price, Dj(p) with This last restriction implies a finite consumer surplus and is useful in guaranteeing a solution to the problem. These demands could be assumed to have been derived from the final demands for that particular use. The resources are not perfectly substitutable between uses. Some resources may be better suited for particular uses, such as petroleum products as fuel for automobiles, whereas other contributions may be more problematic, for example, running an automobile on coal. These differences between resources are summed up into a conversion cost, vij>0, which is the cost of converting a unit of resource I for demand j. Resource units are equivalent for particular demands once the demand specific conversion cost has been expended. We define the net cost of supplying resource I to demand j as wij ≡ ci + vij.
The social planner chooses the quantity of each resource supplied to each demand. We denote by dij(t) the quantity of resource i supplied to demand j at time t. The problem is to determine the resource allocation that maximizes the present value of net social benefit. Given a discount rate r>0, this can be posed as the following optimal control problem:
Choose dij(t) for i Є R and j Є U to maximize
(1)
subject to
dij(t) ≥ 0, qi(t) ≥ 0 for i Є R and j Є U(2)
and
for i Є R.(3).
The state variable qi(t) is the residual stock of resource i over time. The first bracketed term of (1) is the standard consumer surplus of the resources and the second term is the producer surplus. The current value Hamiltonian for the above problem is given by
where λi(t) ≥0 has the standard interpretation as the royalty of resource i.
The solution is defined in terms of optimal price paths as functions of time. Let the price of the resource input for demand j be The necessary conditions for a solution are then
for i Є R.(4)
(5)
pj(t) ≤ wij + λi(t) (if then dij(t)=0) for i Є R and j Є U(6)
and
for i Є R.(7)
We can now develop the following propositions:
PROPOSITION 1. There exists a unique optimal solution to program (1)-(3) and the necessary conditions (4)-(7) are also sufficient.
PROOF: See the Appendix of Chakravorty and Krulce (1994).
The basic principle of resource use, proved by the following proposition, is that the resource that is available at the lowest price (net cost plus royalty) is always used for each demand.
PROPOSITION 2. The price (net cost plus royalty) of a resource that is supplied for a given demand is no more that of any alternative resource.
PROOF: Suppose that daj(t)>0 for some aЄR, jЄU and tЄ(t0,∞). Then from (6) waj+ λa(t) = pj(t) ≤ wij+ + λi(t) for iЄR. Q.E.D.
Consider the resource royalty λi(t). Solving (5) produces the familiar Hotelling equation
for i Є R(8)
which states that royalty rises at the rate of interest. Condition (8) also implies that the royalties of all resources are ordered. Based on this ordering, we write λa < λb to mean λa(t) < λb(t) for all t Є (t0,∞). It may also be the case that the royalties of two resources are the same. As shown by the following proposition, this must be the case if two resources ever simultaneously supply the same demand.
PROPOSITION 3. Two resources simultaneously supplying the same demand have the same royalty and net cost for that demand.
PROOF. Let daj(t)>0 and dbj(t)>0 for some a,bЄ R, j Є U, and t Є I where I is an open interval. Then from (6),
for t Є I.(9)
Differentiating, we get for t Є I. Then from (5) and (8), λa=λb. That is, the royalties are the same. Combining with (9) yields waj = wbj. That is, the net costs are equal. Q.E.D.
Before proceeding, we prove the useful result that all resources approach exhaustion in the limit.
LEMMA. for i ЄR.
PROOF. Pick a ЄR and suppose that λa(t0)=0. Then from (8), λa(t)=0 and so from (6), p1(t)≤wa1. Since demand is positive and downward sloping,
Thus
so there exists bЄ R such that From (4), and so eventually qb(t) will become negative which contradicts (2). Thus the supposition is false and so λa(t0)>0. Combining (7) and (8) yields which since λa(t0)>0 implies that Since a was arbitrary, then for for i ЄR. Q.E.D.
In the standard Hotelling model with a single demand, resource royalties are ordered by cost: the resource with the lowest cost has the highest royalty. With heterogenous demand, the ordering of resource royalties is more problematic since there is not necessarily an ordering of costs among resources. One resource may be cheaper for one demand and more costly for another demand when compared to other resources. The following definitions relate three different types of cost orderings that may occur:
DEFINITION. Resource a ЄR has absolute advantage relative to resource b ЄR in use j ЄU if waj < wbj, some j Є U.
DEFINITION. Resource aЄR is more efficient (less efficient) than resource b ЄR if waj < wbj (waj > wbj) for all j Є U.
DEFINITION. Resource a,bЄR are the same resource class if wbj – waj = k for all jЄ U and some constant k. Furthermore, if k>0 (k=0, k<0) then resource a is a higher grade (same grade, lower grade) of the resource class than resource b.
A resource that is more efficient is strictly cheaper for all demands. Thus efficiency implies Ricardian absolute advantage in all uses. By resource class we mean intuitively that we can classify different resources as different types of “coal”, “oil”, “gas”, etc. where these classifications imply equivalence among demands. The difference between resources within any class is only cost – higher grade resources have a lower cost and this difference in cost is the same for all demands. Note that this classification is based on the economic properties of the resource, not its chemical properties. It may be that resources that are economically the same have completely different chemical compositions.
The next two propositions generalize the principle of cost-ordered royalties.
PROPOSITION 4. More efficient resources have a larger royalty.
PROOF: Let waj<wbj for resources a,b ЄR and all demands j Є U. Suppose that λa ≤ λb. Then from (6), pj(t) ≤ waj + λa(t) <wbj + λb(t) for all t Є (t0,∞). Thus dbj(t)=0 for all t Є (t0,∞) and j Є U; resource b is never extracted for any demand. Since this contradicts the Lemma, the supposition is false and thus λa > λb. Q.E.D.
PROPOSITION 5. Higher grade resources have a larger royalty.
PROOF: If resource a Є R is a higher grade of the same resource class as resource b Є R, then by definition, wbj – waj = k >0 for j Є U which implies that waj< wbj for j Є U. Then from Proposition 4, resource a has a larger royalty than resource b. Q.E.D.
With homogenous demand, the Herfindahl principle states that resources are extracted, sequentially, in order of cost. The following two propositions generalize this principle to show that the use of resources is always in order of net cost and that resources are extracted by decreasing grade.
PROPOSITION 6. Resources are supplied for a given demand in order of increasing net cost.
PROOF. We show that if a resource is supplied for a given demand then a lower net cost resource will not subsequently be supplied for that demand. Thus resources supplied for a given demand must be in order of increasing net cost.
Let daj(t1)>0 and waj>wbj for resources a,bЄ R and demand j ЄU at time t1 Є (t0,∞). Then from (6),
(10)
which since waj > wbj implies that λa < λb. Then from (5), and so the left hand side of (10) increases more slowly than the right hand. Thus
for all t Є (t1,∞).
Then from (6), pj(t) ≤ λa(t) + waj < λb(t) + wbj for all t Є (t1,∞) and so dbj(t)=0 for all t Є (t1,∞). Q.E.D.
Note that Proposition 6 does not say that all resources will be supplied for each demand but that of those resources that are supplied, their use will be in strict order of increasing net cost. In the special case of a single demand, resource use is identical to resource extraction and Proposition 6 reduces to the Hefindahl principle.
PROPOSITION 7. Resources of the same resource class are extracted in order of decreasing grade.
PROOF. We show that if one resource is being extracted, than a higher grade of the same resource will not be subsequently be extracted. Thus resources of the same class are extracted in order of decreasing grade.
Let and for resources a,b Є R at time t1 Є (t0,∞) where resource b is a higher grade of the same resource class as resource a. By the last inequality, from (4) there exists c Є U such that dac(t1)>0. Then from (6),
(11)
which since wbc<wac, from the definition of higher grade, implies that λa(t1) < λb(t1). Then from (5), , the left hand side of (11) increases more slowly than the right hand side, and so wac + λa(t) < wbc + λb(t) for all t Є (t1,∞), which since wbj – waj is constant for all j Є U (from the definition of resource class) implies that waj + λa(t) < wbj + λb(t) for all t Є (t1,∞) and j Є U. Combining with (6) yields pj(t) ≤ waj + λa(t) < wbj + λb(t) for all t Є (t1,∞) and j Є U which implies that dbj(t)=0 for all t Є (t1, ,∞) and j Є U. Then from (4),
for all t Є (t1, ,∞). Q.E.D.
Note that if there is a single resource class, all demands can be aggregated into one composite demand and Proposition 7 reduces to the Herfindahl Principle. Since Proposition 7 demonstrates that deposits within a resource class will be extracted in strict order of grade, we can aggregate resource grades and consider the resulting composite resource that has an extraction cost function that increases with cumulative extraction. This provides a microeconomic foundation for resources with rising, cumulative extraction cost functions, used widely in the literature (e.g, Heal, 1974).
Since demand is positive at all prices, there will always be some resource available for each demand. The following proposition provides a condition under which all resources except one will be exhausted.