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Resource extraction of imperfect substitutes
Resource extraction of imperfect substitutes Franz Wirl
The recent literature about exhaustible resource extraction emphasized the analysis of perfect substitutes (‘backstop’), but overlooked the important case of imperfect substitutes. This paper introduces imperfect substitutes and investigates whether properties, which are robust in one commodity case, carry over to the case of limited substitution. In particular, the proposition ofstrict sequential exploitation of reservoirs of a certain resource from the cheapest to the more costly grades and the validity of declining extraction are investigatedfor the more general case of imperfect substitutes. It will be shown that these properties are not robust against variations in the assumption of the degree of competition of the resource types contained in the diflerence pools. K~.Lww~.~:Exhaustible
resources;
Imperfect
This paper extends the conventional paradigm on extraction policies between a resource and a perfect substitute to the case of imperfect substitutes. The (perfect) substitute is usually called backstop, compare Nordhaus [S], Heal [2], Stiglitz and Dasgupta [14, 15, 161, Kemp and Long [6]. This paper investigates the validity of standard propositions from the field of the theory on exhaustible resources (the Symposium of the Review of Economic Studies [17]; Dasgupta and Heal [l]; and Kemp and Long [S] provide comprehensive surveys), namely, declining extraction policies and sequential exploitation from the cheaper exploitable deposits to the less easily accessible and more costly grades. The last property was proven by Herfindahl [3] and reconsidered by Kemp and Long [S]. Moreover, this result is utilized in many resource models, eg Pindyck [9, lo] in the implicit form of a cost function which is negatively inclined to reserves. Robson [ 1l] derives joint extraction strategies, however, in a stochastic framework.
The author is with the Technische Universitlt Wien, Institut fiir Energiewirtschaft, Gt$hausstrasse 27-29, A-1040 Wien, Austria. Final manuscript
242
received
12 November
1987.
substitutes;
Resource
extraction
The first characteristic of monotonically declining extraction paths is common to all those theoretical investigations which suppose an autonomous demand and monopolistic or competitive markets. The current investigation is limited to these two market structures.
Optimal extraction of identical products Consider a firm which extracts an identical product from two different reservoirs (for a treatment of common pool problems see Khalatbari [7]). The only difference is that the extraction from the second pool is more costly than from the first. For simplicity assume that the extraction from the first pool is costless and that the production from the second reservoir involves constant marginal (or average) production cost. Furthermore, the production facilities are technically independent so that the output can be independently chosen for each facility.’ Typical examples for this description of the production technology are: extraction of oil of similar quality from different fields, mining of coal surface and in situ. However, the extraction of ’ The assumption of separate extraction excludes the interesting of the depletion of crude oil and associated gas.
0140-9883/88/030242-07
$03.00
0
1988 Butterworth
& Co (Publishers)
case
Ltd
Resource extraction qf impeyfict substitutes: F. Wirl
conventional oil and heavy oil or tar sands might better fit the model given later. The production of substitutes like oil and gas or of different metals from independent reserves are typical examples of imperfect substitutes and hence do not comply with the assumptions of this section. Hence, a comparison between the results of this section and its follow-up indicates the type of bias which is introduced if one is too generous about the differences of substitution. This generous but simple assumption is often found in studies about the world energy and oil market where different commodities: coal, liquified coal, natural gas, shale oil, tar sands, breeder reactors are treated as perfect substitutes. Notice, that most of these products depend on an exhaustible resource basis. First suppose that the exploiting firm or a cartel controls a significant fraction of the market and faces a conventional, downward sloping inverse demand function J which depends on aggregate supply. P = f(x + Y) Furthermore assume, that the choke price p = f(0)> c is finite and that the monopolist’s revenue function (and hence the profit function due to the assumption of linear costs) is concave. This ensures a unique solution to the firm’s profit maximization calculus: T
max x,y.T
e-rW)x(r) s
+ (tit) -
cM4 fft
(2)
o
subject to Equation (1) and the state equations which trace cumulative production X(t), Y(t) until the depletion manifold is reached over an optimal production horizon T: k(t) = x(t)
X(0) = 0, X(T)
Y(t) = Y(C)
Y(O)=O,
Y(T)=
= R’
(3)
R2
(4)
x(t) > 0
(5)
y(t) 2 0
(6)
Thereby r denotes the discount rate and R1 and R2 quantify the reserves available at the beginning of the venture. The inequalities (5) and (6) state that the extractions must be non-negative. The optimal policy of the above control problem (lH6) shows the typical characteristics of resource models. These properties are summarized in Proposition 1 and in Figure 1. Since the proof is straightforward and applies well known conditions it is moved to Appendix 1. Proposition
which
I. The aggregate supply of the commodity is extracted by a monopolistic firm declines
ENERGY
ECONOMICS
July 1988
\
\
\ Y-\
Srn
Figure 1. Monopolistic tutes.
\
\
\
\
\
)
Tm
t
resource extraction of perfect substi-
monotonically and continuously to zero. However, the extraction moves sequentially and discontinuously from the exploitation of the first resource pool to the second and more costly extractable reservoir. Figure 1 provides a graphical representation of the result of strictly sequential exploitation and S” and T” mark the depletion dates. The index m is used to label the monopolistic strategies. However, the qualitative nature of the extraction policy is similar to a competitive market. The price is exogenously given to small firms so that a competitive equilibrium can only hold along the singular arc. This result is well known and some of the consequences are summarized in Proposition 2. Proposition 2. The exploitation of the resource pools is strictly sequential from the low cost resource to the more costly extraction of the second reservoir. The price rise in percentages is equal to the rate of interest as long as the first pool is extracted, but is slowed down at the moment the second reservoir becomes productive because then only the profit margin rises exponentially. Hence, the production - aggregate and from each pool - declines monotonically and continuously. The proof of Proposition 2 is omitted because all these properties are well established, eg Hotelling [4] and Kemp and Long [S].
The exploitation
of imperfect substitutes
The assumptions about the technology of production are as given earlier. However, the two extractable commodities which are the content of the two seperated pools are only imperfect substitutes from the perspective of the consumers. More precisely, assume
243
‘Resource extraction
of imperfect substitutes:
that the demand is governed demand functions:
F. Wirl
by a system
The Hamiltonian
of inverse
P1= f’(xv Y)
(7)
Y)
(8)
P2
=f2k
of this control
which satisfies the following conditions: f is twice continuously differentiable with negative first derivatives such that the determinant of the Jacobi matrix is positive. Additionally assume that the profit function,
H = X(P’ + ~1) + (P2 - c +
MR’(x,y):=xp:+p’+yp;=
(9)
which is implied from the demand relations (7) and (8), is concave and that the marginal profit (and hence the marginal revenues due to linear costs) from each commodity are negatively affected by a simultaneous increase in the sales of the competing substitute, ie n,.. < 0. Furthermore, suppose that the backstop (or choke) prices p’, p2 are finite. Let 3’(y) < b’ denote the conditional backstop price, ie that price if the first commodity is withheld from the market but y units from the second good are available; similarly d2. The relations (7) and (8) as well as the properties of the first derivatives stem from a conventional demand system-quantities depending on prices-with negative own- and positive cross-price elasticities. If the ownprice elasticities dominate, ie the product from the own-price elasticities exceeds the product from the cross-price elasticities then such a demand system can be inverted to derive Equations (7) and (8). The above properties of the first derivatives are then a consequence from theorems about diffeomorphisms (eg Rudin [12]). Observe, that the magnitude of the determinant of the Jacobi matrix of (7) and (8) measures the degree of substitutability in such a way that a smaller determinant indicates closer competition among the products. The profit maximizing monopolistic industry will now attempt to solve the following optimal control problem if the supply satisfies the technological assumptions of the previous section.
r z
max X.Y.T
e-“[p’(t)x(t)
Jo
+ (p2(t) - c)yjQ] dt
(11)
~21~
-p,
(12)
=c-p2
(13)
b1=
rP1
(14)
b2 =
w2
(15)
H(T) = 0
(16)
If oil and gas are utilized as examples for x and y then the optimality condition (12) requires that the marginal revenues from oil trade (xp, + p’) minus the (marginal) loss in profits from gas sales due to a decline in the price of gas (yp,) must be equal to the resource shadow price (- pl). A similar interpretation is applicable to gas except that the marginal revenues including the effect on oil profits must balance the costs plus the resource shadow price of gas resources. It is immediately obvious from the inspection of Equations (12) and (13) that these relations allow for joint output from both reservoirs for some time. Hence the result of the previous section about strict sequential exploitation will not be applicable to the current framework. Dividing the relations (12) and (13) establishes that the optimality conditions map the optimal extraction rates for joint output into a straight line in the coordinates of the marginal profits (xx, z,,). This is shown in Figure 2 with the bold lines assuming that x, the costless extractable resource, is depleted first. The same rule applies for the resource prices in competitive markets whether the resource based goods are perfectly or imperfectly substitutable. The result is well known as Hotelling’s rule: @’ = rp’
(17)
p2 = r(p2 - c)
(18)
(10) I
subject to the demand relations (7) and (8), the state equations (3) and (4) and the non-negativity constraints (5) and (6). ‘5 describes the ultimate date of depletion, ie T = max(S, T), where S,Tdenote the depletion dates of the reservoirs from x and y respectively. The indices c, m will be again used to discriminate between the competitive and the monopolistic solution.
244
is defined
where the arguments are suppressed to abbreviate the expressions. Necessary (and sufficient) conditions for an optimal exploitation strategy (ie interior paths) are:2
MR2(x, y):=yp,2+p2+xp; 7L:=p’x+(p2-c)y
problem
as:
so that Equations (17) and (18) do not differ qualitatively from the results earlier. Equations (17) and (18)
’ Low case indices are utilized derivatives, eg
in the following
to indicate
partial
pi= ap'iax except for the shadow
prices pi.
ENERGY
ECONOMICS
July 1988
Resource e.rtraction
of imprrfkct substitutes:
F. Il’irl
18 16 14 12 10 8 6 4 2 0 10
0
20
30
50
40
60
Ttme
Figure 3. Example of a monopolistic imperfect
i;’ Figure 2. Optimal
price policies of monopoly tion drawn in different coordinate systems.
7-J
and competi-
hold if both resources are extracted, otherwise the (conditional) backstop price is applied at the dates where one commodity is not produced or is already depleted. This result can be shown as a straight line, the coordinates ‘prices’ at the dates of joint output if the costs c are negligible. This is shown also in Figure 2 and indicated by the broken curve, where, however, the coordinates are different from the monopolistic solution. As already indicated, the major conclusions from this framework are that the earlier propositions must, in general, be rejected if the considered resources are only imperfectly substitutable. These refutations are established by the use of counter examples and are surprising because of the similar nature of the optimality conditions. These examples, which are documented in Appendix 2, suppose a linear demand system which satisfies the assumptions made at the beginning of this section. Moreover, the assumed backstop prices of the two products are either identical or similar. Hence, a fairly simple framework and not a specific or complicated artificial construct is sufficient to reject the extension of Propositions 1 and 2. In particular the following is found. Extraction of imperfectly substitutable resources characterized additionally by different extraction costs is in general neither strictly sequential nor monotonically. This applies to both market regimes: monopoly and competition. This result deviates substantially from those models which predict a discontinuous transition from the use of one resource to the other. Figures 3 and 4 show the
ENERGY
ECONOMICS
July 1988
depletion policy for
substitutes.
results from the examples which support the refutation of the propositions. However, one can construct sufficient conditions that monotonically declining extraction rates are still applicable. In order to simplify these conditions suppose that c is small. Then, the assumption that the own-price elasticity exceeds the crossprice elasticity globally and for each demand would ensure that the extractions must decline monotonically in competitive markets. In the case of monopolistic allocations the supposition that each marginal revenue function is more elastic in respect to own-than to cross-production, ie aln MR’
x
MR’ (19)
aIn y
aln MR2
dln MR2
din y
<
dn
(20)
x
24 22 20 18 16 14 12 10 8 6 4 2 0 0
10
20
30
40
50
60
Time
Figure 4. Example of extraction rates of imperfect substitutes in competitive
markets.
245
Resourc~c~estraction
suhstitutc~s: F. Wirl
of‘ inpzrftict
ensures monotonically declining productions. These inequalities (19) and (20) are established by differentiating (12) and (13) in respect to time and requiring that s < 0, J’ < 0. This differentiation reveals immediately that both extractions cannot rise simultaneously. Although the above conclusions reject strict sequential exploitation, the graphs may lead to the conjecture that the result can be saved in a moderated form, that the costless extractable resource is always first depleted. However, this is in general not true. The sequence of depletion depends on all model parameters and not solely on costs, ie it may be optimal to conserve the low cost resource pool and to deplete the costly reservoir at first. Moreover, the sequence of depletion may be different for the two market regimes, but otherwise identical demand/supply characteristics. This conclusion is graphically shown in Figure 5 which draws the ratio of the depletion dates, S/T, uis a vis the ratio of initial reserves, R’/R’, for both market structures.
Competition from a synthetic but imperfect substitute The papers of Kemp and Long [6], Heal [2] and Salant [13], but also Hoe1 [4] and of some other authors consider a market where a resource exploiting monopoly faces competition from a competitive industry which can supply a perfect substitute at cost c. The solution is well known and shown in Figure 6 by broken (resource x’) and solid (backstop, renewable commodity y’) lines. In short, the monopoly chooses a price strategy which rises along the formula that the percentage growth of the marginal revenues must be equal to the rate of discount until the price increases reach the costs of the backstop technology. Then the monopolist will marginally undercut the costs of its competitors until the resource stock is depleted. The competitive industry enters instantaneously at the date 2.0 1.9 1.8 1.:
,
h
I
1.5
l
s
t
and market penetration synfuels: perfect and imperfect substitutes.
of
5’
Figure 6. Resource extraction
of resource depletion and supplies the entire market for the remainder. Now suppose that the knowhow about an imperfect but renewable substitute is already available in the public domain, eg fast breeder reactors should substitute for liquid fuels, and that no economies of scale exist so that a competitive industry will supply this commodity. Then the resource extracting cartel will attempt to maximize the present value of the profit stream n’(x, y) = pl(x, y)x, TT’ concave with negative second derivatives: s max x,s
e-%‘(t)
dt
(21)
s0
subject to the laws of demand (7), the differential equation (3) and the non-negativity constraint (5). However, simultaneously, the competitive industry will supply the substitute if the production costs can be covered by the market price. And if this is possible, it will produce exactly the amount that the market price is equal to the costs: p2 = c. Therefore, y(t) = 0
if p’(x(t), 0) = p2(x) < c
Monopoly
(23)
Hence an intertemporal Nash/Cournot equilibrium (.x!(t), yN(t)) must maximize Equation (21) and satisfy Equations (22) and (23). The optimal depletion strategy is described by the well known condition that the marginal revenues equal the resource shadow price:
A\
7r;=-/l 3.08
0.11
0.:5
0.20
0.24
0.28
R2IR1
Figure 5. Ratio of depletion dates u reserve ratios.
246
(22)
if only the resource is supplied. Otherwise the output from the competitive industry can be computed from the implicit relation: p2(x(t), y(f)) = c if b2(.y) 3 c
-Competlt~on
l.G
t
(24)
0.32
Differentiation of competition
of Equations and solving
ENERGY
(24) and (23) for days for f, j yields: i < 0,
ECONOMICS
July 1988
j> 0. Moreover, it is routine to establish that the synfuel industry penetrates the market smoothly and continuously until it covers the entire market. Figure 6 compares the solutions under different assumptions of a perfect versus an imperfect substitute and the latter case corresponds to the bold lines. The above derived solution is robust and applies also to other market equilibria, eg to a Stackelberg equilibrium. In this case the resource extracting firm replaces y(r) by either zero if b2
7-c;+7ciy’=
-p
(25)
If the left hand side of Equation (22) is concave in x, then the Stackelberg equilibrium is qualitatively of a similar nature to the Nash/Cournot solution (however, the actual production rates differ).
Conclusions This paper reconsidered the economics of intertemporal extraction strategies and criticized the conventional single resource/perfect substitute paradigm. For this purpose the simple models on resource extraction were adapted and extended to two imperfectly substitutable but finite resources. This straightforward extension implied that some essential properties from the conventional theory are not applicable in the more realistic and more general circumstances. In particular it was shown that resource extraction need neither decline monotonically nor will the reservoirs be exploited in a sequential manner from the low cost pool to the high cost grades. If the substitute is a renewable one then the abrupt change in supply pattern typical for perfect substitutes must be altered to a continuous, smooth and gradual transition from the early days of resource use to the later dates of the backstop technology. This solution is similar to the ones which focus on perfect substitutes but slow accumulation of capacities of the synfuel industry. This research may provide the impetus for reconsidering numerous propositions from the traditional theory. Such investigations are not only of theoretical interest but the results in this paper suggest that they may alter our perception of such important commodity markets as the world energy markets. A further application is of an empirical nature: to use this framework, eg the examples which are elaborated in Appendix 2, to project (rational) strategies for pricing both hydrocarbons of OPEC - crude oil and natural gas - in a coherent and simultaneous framework.
ENERGY
ECONOMICS
July 1988
Appendix 1 Extraction of identical resources The Hamiltonian defined,
of the optimal
control
problem
H=px+(p-C)Y+P,X+P,J
(lH6)
is
(Al)
and p,, p2 denote the adjoint variables. Differentiation and the introduction of the marginal revenues denoted by MR _ establishes that MR = -p,
W)
ifx>Oand MR=c-/I,
(A3)
if y > 0. Since the costates
grow exponentially:
h = r/J:
(A4)
P2
(A5)
=
v2
both relations (A2) and (A3) cannot hold simultaneously except for one point in time. Optimal stopping time requires that H(7)=0
(A6)
Hence, extraction from each pool must decline (differentiate (A2) and (A3) in respect to time, use the concavity of the revenues, (A4) and (AS) and the fact that pi
Appendix 2 Examples - linear demand Let p=j+Az P=
(A7)
(P’,P2): P = ($3 p2)‘,iJ =
(x. Y)
describe a linear inverse demand system of type (7) and (8) with the Jacobi matrix A = [aij]. The linear transformations a = - A ‘jj and B = A- I establish the conventional demand system: z=a+Bp
(A@
The maximum
principle
z= -(A+A’)-‘(j,-r+p) l-
=
(0,
cl’,
P =
(P,,
(9HlO)
simplifies
to (A9)
112)
247
e.utraction qf’impe@ct
Resource
substitutes:
F. Wirl
for interior
solution of the monopolist’s strategies. Now that R’ is first depleted, then the condition for depletion (H(r) = 0) implies:
suppose optimal
demand
system:
z=[::]+[-:r; p2(t) = -e
-rlT-t)(pz
_
(_)
(A101
and y(t) = ($ -c + pz(t))/2a,,, t E [Sm, T”], from the conventional monopoly-single commodity problem. This can be used to compute F’(y(Sm)) and from the maximum principle
References 1
2 -~,(S”)=jj’(~(S”))+a,,J?(S”)=p /c,(t) = e-“sm-”
+(a,,
p,(Sm)
+a2,)y(Sm) (Al 1) (Al2) 3
the
shadow price for the first resource pool can be derived. The computations of extractions involve now only (Sm, T”) as unknown parameters for the calculation of the resource shadow prices. The dates of depletion must be chosen such that the integration of (A9) up to S” and T” respectively equals the initial resource stocks R’ and R*. The competitive solution (14) and (15) is completed if appropriate boundary conditions are supposed, eg for the case s’ < T’: p*( T’) z jj’
(Al3)
p’(S’) = p’(p2(S))
(AI4)
that the depletion is smooth and that both depleted.3 Therefore, according to Equation the prices can be computed:
p’(t) = p*(r)
=
t
F’(P2(Q)
t > S”
+
e-r(Tm
reservoirs are (14) and (15)
4
4
5
6
7
8
e-~lS”-~ljjl(P2(~m~)
1 c
Q
y
9 -‘l(f)*
_
C)
6416)
These prices are then inserted into the demand system (A8). The resulting extractions are integrated and the depletion guess (Sm, T”) is modified (eg by a Newton algorithm) until cumulative production up to S” and T” respectively is equal to the initial reserves R’ and R*. The procedure is similar in the case of opposite optimal depletion sequence under both market regimes, ie S > T. The results in Figures 3 and 5 are based on the demand system:
(Al7)
10
11
12 13
in the conventional form (A8). The remaining paraare held constant at: r = 0.05, c = 5.0, R’ = 600, R* = 300 except for the variation of R* in Figure 5. The competitive extractions of Figure 4 are based on a different
meters
3Some authors argue that competitive markets do not guarantee that the proper boundaries are established, eg compare Dasgupta and Heal [I]. However. if the competitive allocation is viewed as the limiting case to a NashiCournot competition between many individual suppliers then the supposed boundary conditions are an integral part of the solution.
Partha
Dasgupta
Theory
and Exhaustible
and
Geoffrey Resources,
M. Heal, Economic Cambridge Econ-
omic Handbooks, 1979. Geoffrey M. Heal, ‘Uncertainty and the optimal supply policy for an exhaustible resource’, in R. S. Pindyck, ed, Advances in the Economics ofEnergy and Resources, Vol 2, JAI Press, 1979, pp 119-147. 0. C. Herfindahl, ‘Depletion and economic theory’, in M. M. Gaffney, ed, Extractive Resources and Taxation, University of Wisconsin Press, 1967, pp 63-69. Michael Hoel, ‘Monopoly resource extractions under the presence of predetermined substitute production’, Journal 0s Economic Theory, Vol 30, 1983, pp 201-2 12. Harold Hotelling, ‘The economics of exhaustible resources’, Journal of Political Economy, Vol 39, 1931, pp 137-175. Murray C. Kemp and Ngo van Long, Exhaustible Resources, Optimality and Trade, North Holland, Amsterdam, 1980. Murray C. Kemp and Ngo van Long, ‘On the development of a substitute for an exhaustible resource’, in W. Eichhorn, ed, Economic 7heory of Natural Resources, Physica, Wiirzburg-Wien, 1982. F. Khalatbari, ‘Market imperfections and optimal rate of depletion of an exhaustible resource’, Economica, Vol 44, 1976, pp 409414. William D. Nordhaus, ‘The allocation of energy resources’, Brookings Papers on Economic Activity, Vol 3, 1973. Robert S. Pindyck, ‘Gains to producers from cartelization’, Review of Economics and Statistics, Vol 60, 1978, pp 238-25 1. Robert S. Pindyck, ‘The optimal exploration and production of nonrenewable resources’, Journal of Political Economy, Vol 86, 1978, pp 841-861. Arthur J. Robson, ‘Sequential exploitation of uncertain deposits of a depletable resource’, Journal of Economic Theory, Vol 21, 1979, pp 88-l IO. Walter Rudin, Principles cf Mathematical Analysis, McGraw Hill, 1953. Stephen W. Salant, ‘Staving off the backstop: dynamic limit pricing with a kinked demand curve, in R. S. Pindyck, ed, Adounw.v in the Economics of’ Energy Resources, JAI Press, Vol 2, pp 187-204.
14
given
24%
(Al8)
_:::I
15
16
17
and
Joseph E. Stiglitz and Partha Dasgupta, ‘Market structure and resource extraction under uncertainty’, Scandinavian Journal ojEconomics, Vol83,198 1, pp 3 18-333. Joseph E. Stiglitz and Partha Dasgupta, ‘Market structure and resource depletion: A contribution to the theory of intertemporal monopolistic competition’, Journal of Economic Theory, Vol 28, 1981, pp 128-164. Joseph E. Stiglitz and Partha Dasgupta, ‘Strategic considerations in invention and innovation: the case of natural resources’, Econometrica, Vol 51, 1983, pp 1439-1448. The Review of’ Economic Studies, Symposium on the Economics of Exhaustible Resources, 1974.
ENERGY
ECONOMICS
July
1988
The recent literature about exhaustible resource extraction emphasized the analysis of perfect substitutes (‘backstop’), but overlooked the important case of imperfect substitutes. This paper introduces imperfect substitutes and investigates whether properties, which are robust in one commodity case, carry over to the case of limited substitution. In particular, the proposition ofstrict sequential exploitation of reservoirs of a certain resource from the cheapest to the more costly grades and the validity of declining extraction are investigatedfor the more general case of imperfect substitutes. It will be shown that these properties are not robust against variations in the assumption of the degree of competition of the resource types contained in the diflerence pools. K~.Lww~.~:Exhaustible
resources;
Imperfect
This paper extends the conventional paradigm on extraction policies between a resource and a perfect substitute to the case of imperfect substitutes. The (perfect) substitute is usually called backstop, compare Nordhaus [S], Heal [2], Stiglitz and Dasgupta [14, 15, 161, Kemp and Long [6]. This paper investigates the validity of standard propositions from the field of the theory on exhaustible resources (the Symposium of the Review of Economic Studies [17]; Dasgupta and Heal [l]; and Kemp and Long [S] provide comprehensive surveys), namely, declining extraction policies and sequential exploitation from the cheaper exploitable deposits to the less easily accessible and more costly grades. The last property was proven by Herfindahl [3] and reconsidered by Kemp and Long [S]. Moreover, this result is utilized in many resource models, eg Pindyck [9, lo] in the implicit form of a cost function which is negatively inclined to reserves. Robson [ 1l] derives joint extraction strategies, however, in a stochastic framework.
The author is with the Technische Universitlt Wien, Institut fiir Energiewirtschaft, Gt$hausstrasse 27-29, A-1040 Wien, Austria. Final manuscript
242
received
12 November
1987.
substitutes;
Resource
extraction
The first characteristic of monotonically declining extraction paths is common to all those theoretical investigations which suppose an autonomous demand and monopolistic or competitive markets. The current investigation is limited to these two market structures.
Optimal extraction of identical products Consider a firm which extracts an identical product from two different reservoirs (for a treatment of common pool problems see Khalatbari [7]). The only difference is that the extraction from the second pool is more costly than from the first. For simplicity assume that the extraction from the first pool is costless and that the production from the second reservoir involves constant marginal (or average) production cost. Furthermore, the production facilities are technically independent so that the output can be independently chosen for each facility.’ Typical examples for this description of the production technology are: extraction of oil of similar quality from different fields, mining of coal surface and in situ. However, the extraction of ’ The assumption of separate extraction excludes the interesting of the depletion of crude oil and associated gas.
0140-9883/88/030242-07
$03.00
0
1988 Butterworth
& Co (Publishers)
case
Ltd
Resource extraction qf impeyfict substitutes: F. Wirl
conventional oil and heavy oil or tar sands might better fit the model given later. The production of substitutes like oil and gas or of different metals from independent reserves are typical examples of imperfect substitutes and hence do not comply with the assumptions of this section. Hence, a comparison between the results of this section and its follow-up indicates the type of bias which is introduced if one is too generous about the differences of substitution. This generous but simple assumption is often found in studies about the world energy and oil market where different commodities: coal, liquified coal, natural gas, shale oil, tar sands, breeder reactors are treated as perfect substitutes. Notice, that most of these products depend on an exhaustible resource basis. First suppose that the exploiting firm or a cartel controls a significant fraction of the market and faces a conventional, downward sloping inverse demand function J which depends on aggregate supply. P = f(x + Y) Furthermore assume, that the choke price p = f(0)> c is finite and that the monopolist’s revenue function (and hence the profit function due to the assumption of linear costs) is concave. This ensures a unique solution to the firm’s profit maximization calculus: T
max x,y.T
e-rW)x(r) s
+ (tit) -
cM4 fft
(2)
o
subject to Equation (1) and the state equations which trace cumulative production X(t), Y(t) until the depletion manifold is reached over an optimal production horizon T: k(t) = x(t)
X(0) = 0, X(T)
Y(t) = Y(C)
Y(O)=O,
Y(T)=
= R’
(3)
R2
(4)
x(t) > 0
(5)
y(t) 2 0
(6)
Thereby r denotes the discount rate and R1 and R2 quantify the reserves available at the beginning of the venture. The inequalities (5) and (6) state that the extractions must be non-negative. The optimal policy of the above control problem (lH6) shows the typical characteristics of resource models. These properties are summarized in Proposition 1 and in Figure 1. Since the proof is straightforward and applies well known conditions it is moved to Appendix 1. Proposition
which
I. The aggregate supply of the commodity is extracted by a monopolistic firm declines
ENERGY
ECONOMICS
July 1988
\
\
\ Y-\
Srn
Figure 1. Monopolistic tutes.
\
\
\
\
\
)
Tm
t
resource extraction of perfect substi-
monotonically and continuously to zero. However, the extraction moves sequentially and discontinuously from the exploitation of the first resource pool to the second and more costly extractable reservoir. Figure 1 provides a graphical representation of the result of strictly sequential exploitation and S” and T” mark the depletion dates. The index m is used to label the monopolistic strategies. However, the qualitative nature of the extraction policy is similar to a competitive market. The price is exogenously given to small firms so that a competitive equilibrium can only hold along the singular arc. This result is well known and some of the consequences are summarized in Proposition 2. Proposition 2. The exploitation of the resource pools is strictly sequential from the low cost resource to the more costly extraction of the second reservoir. The price rise in percentages is equal to the rate of interest as long as the first pool is extracted, but is slowed down at the moment the second reservoir becomes productive because then only the profit margin rises exponentially. Hence, the production - aggregate and from each pool - declines monotonically and continuously. The proof of Proposition 2 is omitted because all these properties are well established, eg Hotelling [4] and Kemp and Long [S].
The exploitation
of imperfect substitutes
The assumptions about the technology of production are as given earlier. However, the two extractable commodities which are the content of the two seperated pools are only imperfect substitutes from the perspective of the consumers. More precisely, assume
243
‘Resource extraction
of imperfect substitutes:
that the demand is governed demand functions:
F. Wirl
by a system
The Hamiltonian
of inverse
P1= f’(xv Y)
(7)
Y)
(8)
P2
=f2k
of this control
which satisfies the following conditions: f is twice continuously differentiable with negative first derivatives such that the determinant of the Jacobi matrix is positive. Additionally assume that the profit function,
H = X(P’ + ~1) + (P2 - c +
MR’(x,y):=xp:+p’+yp;=
(9)
which is implied from the demand relations (7) and (8), is concave and that the marginal profit (and hence the marginal revenues due to linear costs) from each commodity are negatively affected by a simultaneous increase in the sales of the competing substitute, ie n,.. < 0. Furthermore, suppose that the backstop (or choke) prices p’, p2 are finite. Let 3’(y) < b’ denote the conditional backstop price, ie that price if the first commodity is withheld from the market but y units from the second good are available; similarly d2. The relations (7) and (8) as well as the properties of the first derivatives stem from a conventional demand system-quantities depending on prices-with negative own- and positive cross-price elasticities. If the ownprice elasticities dominate, ie the product from the own-price elasticities exceeds the product from the cross-price elasticities then such a demand system can be inverted to derive Equations (7) and (8). The above properties of the first derivatives are then a consequence from theorems about diffeomorphisms (eg Rudin [12]). Observe, that the magnitude of the determinant of the Jacobi matrix of (7) and (8) measures the degree of substitutability in such a way that a smaller determinant indicates closer competition among the products. The profit maximizing monopolistic industry will now attempt to solve the following optimal control problem if the supply satisfies the technological assumptions of the previous section.
r z
max X.Y.T
e-“[p’(t)x(t)
Jo
+ (p2(t) - c)yjQ] dt
(11)
~21~
-p,
(12)
=c-p2
(13)
b1=
rP1
(14)
b2 =
w2
(15)
H(T) = 0
(16)
If oil and gas are utilized as examples for x and y then the optimality condition (12) requires that the marginal revenues from oil trade (xp, + p’) minus the (marginal) loss in profits from gas sales due to a decline in the price of gas (yp,) must be equal to the resource shadow price (- pl). A similar interpretation is applicable to gas except that the marginal revenues including the effect on oil profits must balance the costs plus the resource shadow price of gas resources. It is immediately obvious from the inspection of Equations (12) and (13) that these relations allow for joint output from both reservoirs for some time. Hence the result of the previous section about strict sequential exploitation will not be applicable to the current framework. Dividing the relations (12) and (13) establishes that the optimality conditions map the optimal extraction rates for joint output into a straight line in the coordinates of the marginal profits (xx, z,,). This is shown in Figure 2 with the bold lines assuming that x, the costless extractable resource, is depleted first. The same rule applies for the resource prices in competitive markets whether the resource based goods are perfectly or imperfectly substitutable. The result is well known as Hotelling’s rule: @’ = rp’
(17)
p2 = r(p2 - c)
(18)
(10) I
subject to the demand relations (7) and (8), the state equations (3) and (4) and the non-negativity constraints (5) and (6). ‘5 describes the ultimate date of depletion, ie T = max(S, T), where S,Tdenote the depletion dates of the reservoirs from x and y respectively. The indices c, m will be again used to discriminate between the competitive and the monopolistic solution.
244
is defined
where the arguments are suppressed to abbreviate the expressions. Necessary (and sufficient) conditions for an optimal exploitation strategy (ie interior paths) are:2
MR2(x, y):=yp,2+p2+xp; 7L:=p’x+(p2-c)y
problem
as:
so that Equations (17) and (18) do not differ qualitatively from the results earlier. Equations (17) and (18)
’ Low case indices are utilized derivatives, eg
in the following
to indicate
partial
pi= ap'iax except for the shadow
prices pi.
ENERGY
ECONOMICS
July 1988
Resource e.rtraction
of imprrfkct substitutes:
F. Il’irl
18 16 14 12 10 8 6 4 2 0 10
0
20
30
50
40
60
Ttme
Figure 3. Example of a monopolistic imperfect
i;’ Figure 2. Optimal
price policies of monopoly tion drawn in different coordinate systems.
7-J
and competi-
hold if both resources are extracted, otherwise the (conditional) backstop price is applied at the dates where one commodity is not produced or is already depleted. This result can be shown as a straight line, the coordinates ‘prices’ at the dates of joint output if the costs c are negligible. This is shown also in Figure 2 and indicated by the broken curve, where, however, the coordinates are different from the monopolistic solution. As already indicated, the major conclusions from this framework are that the earlier propositions must, in general, be rejected if the considered resources are only imperfectly substitutable. These refutations are established by the use of counter examples and are surprising because of the similar nature of the optimality conditions. These examples, which are documented in Appendix 2, suppose a linear demand system which satisfies the assumptions made at the beginning of this section. Moreover, the assumed backstop prices of the two products are either identical or similar. Hence, a fairly simple framework and not a specific or complicated artificial construct is sufficient to reject the extension of Propositions 1 and 2. In particular the following is found. Extraction of imperfectly substitutable resources characterized additionally by different extraction costs is in general neither strictly sequential nor monotonically. This applies to both market regimes: monopoly and competition. This result deviates substantially from those models which predict a discontinuous transition from the use of one resource to the other. Figures 3 and 4 show the
ENERGY
ECONOMICS
July 1988
depletion policy for
substitutes.
results from the examples which support the refutation of the propositions. However, one can construct sufficient conditions that monotonically declining extraction rates are still applicable. In order to simplify these conditions suppose that c is small. Then, the assumption that the own-price elasticity exceeds the crossprice elasticity globally and for each demand would ensure that the extractions must decline monotonically in competitive markets. In the case of monopolistic allocations the supposition that each marginal revenue function is more elastic in respect to own-than to cross-production, ie aln MR’
x
MR’ (19)
aIn y
aln MR2
dln MR2
din y
<
dn
(20)
x
24 22 20 18 16 14 12 10 8 6 4 2 0 0
10
20
30
40
50
60
Time
Figure 4. Example of extraction rates of imperfect substitutes in competitive
markets.
245
Resourc~c~estraction
suhstitutc~s: F. Wirl
of‘ inpzrftict
ensures monotonically declining productions. These inequalities (19) and (20) are established by differentiating (12) and (13) in respect to time and requiring that s < 0, J’ < 0. This differentiation reveals immediately that both extractions cannot rise simultaneously. Although the above conclusions reject strict sequential exploitation, the graphs may lead to the conjecture that the result can be saved in a moderated form, that the costless extractable resource is always first depleted. However, this is in general not true. The sequence of depletion depends on all model parameters and not solely on costs, ie it may be optimal to conserve the low cost resource pool and to deplete the costly reservoir at first. Moreover, the sequence of depletion may be different for the two market regimes, but otherwise identical demand/supply characteristics. This conclusion is graphically shown in Figure 5 which draws the ratio of the depletion dates, S/T, uis a vis the ratio of initial reserves, R’/R’, for both market structures.
Competition from a synthetic but imperfect substitute The papers of Kemp and Long [6], Heal [2] and Salant [13], but also Hoe1 [4] and of some other authors consider a market where a resource exploiting monopoly faces competition from a competitive industry which can supply a perfect substitute at cost c. The solution is well known and shown in Figure 6 by broken (resource x’) and solid (backstop, renewable commodity y’) lines. In short, the monopoly chooses a price strategy which rises along the formula that the percentage growth of the marginal revenues must be equal to the rate of discount until the price increases reach the costs of the backstop technology. Then the monopolist will marginally undercut the costs of its competitors until the resource stock is depleted. The competitive industry enters instantaneously at the date 2.0 1.9 1.8 1.:
,
h
I
1.5
l
s
t
and market penetration synfuels: perfect and imperfect substitutes.
of
5’
Figure 6. Resource extraction
of resource depletion and supplies the entire market for the remainder. Now suppose that the knowhow about an imperfect but renewable substitute is already available in the public domain, eg fast breeder reactors should substitute for liquid fuels, and that no economies of scale exist so that a competitive industry will supply this commodity. Then the resource extracting cartel will attempt to maximize the present value of the profit stream n’(x, y) = pl(x, y)x, TT’ concave with negative second derivatives: s max x,s
e-%‘(t)
dt
(21)
s0
subject to the laws of demand (7), the differential equation (3) and the non-negativity constraint (5). However, simultaneously, the competitive industry will supply the substitute if the production costs can be covered by the market price. And if this is possible, it will produce exactly the amount that the market price is equal to the costs: p2 = c. Therefore, y(t) = 0
if p’(x(t), 0) = p2(x) < c
Monopoly
(23)
Hence an intertemporal Nash/Cournot equilibrium (.x!(t), yN(t)) must maximize Equation (21) and satisfy Equations (22) and (23). The optimal depletion strategy is described by the well known condition that the marginal revenues equal the resource shadow price:
A\
7r;=-/l 3.08
0.11
0.:5
0.20
0.24
0.28
R2IR1
Figure 5. Ratio of depletion dates u reserve ratios.
246
(22)
if only the resource is supplied. Otherwise the output from the competitive industry can be computed from the implicit relation: p2(x(t), y(f)) = c if b2(.y) 3 c
-Competlt~on
l.G
t
(24)
0.32
Differentiation of competition
of Equations and solving
ENERGY
(24) and (23) for days for f, j yields: i < 0,
ECONOMICS
July 1988
j> 0. Moreover, it is routine to establish that the synfuel industry penetrates the market smoothly and continuously until it covers the entire market. Figure 6 compares the solutions under different assumptions of a perfect versus an imperfect substitute and the latter case corresponds to the bold lines. The above derived solution is robust and applies also to other market equilibria, eg to a Stackelberg equilibrium. In this case the resource extracting firm replaces y(r) by either zero if b2
7-c;+7ciy’=
-p
(25)
If the left hand side of Equation (22) is concave in x, then the Stackelberg equilibrium is qualitatively of a similar nature to the Nash/Cournot solution (however, the actual production rates differ).
Conclusions This paper reconsidered the economics of intertemporal extraction strategies and criticized the conventional single resource/perfect substitute paradigm. For this purpose the simple models on resource extraction were adapted and extended to two imperfectly substitutable but finite resources. This straightforward extension implied that some essential properties from the conventional theory are not applicable in the more realistic and more general circumstances. In particular it was shown that resource extraction need neither decline monotonically nor will the reservoirs be exploited in a sequential manner from the low cost pool to the high cost grades. If the substitute is a renewable one then the abrupt change in supply pattern typical for perfect substitutes must be altered to a continuous, smooth and gradual transition from the early days of resource use to the later dates of the backstop technology. This solution is similar to the ones which focus on perfect substitutes but slow accumulation of capacities of the synfuel industry. This research may provide the impetus for reconsidering numerous propositions from the traditional theory. Such investigations are not only of theoretical interest but the results in this paper suggest that they may alter our perception of such important commodity markets as the world energy markets. A further application is of an empirical nature: to use this framework, eg the examples which are elaborated in Appendix 2, to project (rational) strategies for pricing both hydrocarbons of OPEC - crude oil and natural gas - in a coherent and simultaneous framework.
ENERGY
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July 1988
Appendix 1 Extraction of identical resources The Hamiltonian defined,
of the optimal
control
problem
H=px+(p-C)Y+P,X+P,J
(lH6)
is
(Al)
and p,, p2 denote the adjoint variables. Differentiation and the introduction of the marginal revenues denoted by MR _ establishes that MR = -p,
W)
ifx>Oand MR=c-/I,
(A3)
if y > 0. Since the costates
grow exponentially:
h = r/J:
(A4)
P2
(A5)
=
v2
both relations (A2) and (A3) cannot hold simultaneously except for one point in time. Optimal stopping time requires that H(7)=0
(A6)
Hence, extraction from each pool must decline (differentiate (A2) and (A3) in respect to time, use the concavity of the revenues, (A4) and (AS) and the fact that pi
Appendix 2 Examples - linear demand Let p=j+Az P=
(A7)
(P’,P2): P = ($3 p2)‘,iJ =
(x. Y)
describe a linear inverse demand system of type (7) and (8) with the Jacobi matrix A = [aij]. The linear transformations a = - A ‘jj and B = A- I establish the conventional demand system: z=a+Bp
(A@
The maximum
principle
z= -(A+A’)-‘(j,-r+p) l-
=
(0,
cl’,
P =
(P,,
(9HlO)
simplifies
to (A9)
112)
247
e.utraction qf’impe@ct
Resource
substitutes:
F. Wirl
for interior
solution of the monopolist’s strategies. Now that R’ is first depleted, then the condition for depletion (H(r) = 0) implies:
suppose optimal
demand
system:
z=[::]+[-:r; p2(t) = -e
-rlT-t)(pz
_
(_)
(A101
and y(t) = ($ -c + pz(t))/2a,,, t E [Sm, T”], from the conventional monopoly-single commodity problem. This can be used to compute F’(y(Sm)) and from the maximum principle
References 1
2 -~,(S”)=jj’(~(S”))+a,,J?(S”)=p /c,(t) = e-“sm-”
+(a,,
p,(Sm)
+a2,)y(Sm) (Al 1) (Al2) 3
the
shadow price for the first resource pool can be derived. The computations of extractions involve now only (Sm, T”) as unknown parameters for the calculation of the resource shadow prices. The dates of depletion must be chosen such that the integration of (A9) up to S” and T” respectively equals the initial resource stocks R’ and R*. The competitive solution (14) and (15) is completed if appropriate boundary conditions are supposed, eg for the case s’ < T’: p*( T’) z jj’
(Al3)
p’(S’) = p’(p2(S))
(AI4)
that the depletion is smooth and that both depleted.3 Therefore, according to Equation the prices can be computed:
p’(t) = p*(r)
=
t
F’(P2(Q)
t > S”
+
e-r(Tm
reservoirs are (14) and (15)
4
4
5
6
7
8
e-~lS”-~ljjl(P2(~m~)
1 c
Q
y
9 -‘l(f)*
_
C)
6416)
These prices are then inserted into the demand system (A8). The resulting extractions are integrated and the depletion guess (Sm, T”) is modified (eg by a Newton algorithm) until cumulative production up to S” and T” respectively is equal to the initial reserves R’ and R*. The procedure is similar in the case of opposite optimal depletion sequence under both market regimes, ie S > T. The results in Figures 3 and 5 are based on the demand system:
(Al7)
10
11
12 13
in the conventional form (A8). The remaining paraare held constant at: r = 0.05, c = 5.0, R’ = 600, R* = 300 except for the variation of R* in Figure 5. The competitive extractions of Figure 4 are based on a different
meters
3Some authors argue that competitive markets do not guarantee that the proper boundaries are established, eg compare Dasgupta and Heal [I]. However. if the competitive allocation is viewed as the limiting case to a NashiCournot competition between many individual suppliers then the supposed boundary conditions are an integral part of the solution.
Partha
Dasgupta
Theory
and Exhaustible
and
Geoffrey Resources,
M. Heal, Economic Cambridge Econ-
omic Handbooks, 1979. Geoffrey M. Heal, ‘Uncertainty and the optimal supply policy for an exhaustible resource’, in R. S. Pindyck, ed, Advances in the Economics ofEnergy and Resources, Vol 2, JAI Press, 1979, pp 119-147. 0. C. Herfindahl, ‘Depletion and economic theory’, in M. M. Gaffney, ed, Extractive Resources and Taxation, University of Wisconsin Press, 1967, pp 63-69. Michael Hoel, ‘Monopoly resource extractions under the presence of predetermined substitute production’, Journal 0s Economic Theory, Vol 30, 1983, pp 201-2 12. Harold Hotelling, ‘The economics of exhaustible resources’, Journal of Political Economy, Vol 39, 1931, pp 137-175. Murray C. Kemp and Ngo van Long, Exhaustible Resources, Optimality and Trade, North Holland, Amsterdam, 1980. Murray C. Kemp and Ngo van Long, ‘On the development of a substitute for an exhaustible resource’, in W. Eichhorn, ed, Economic 7heory of Natural Resources, Physica, Wiirzburg-Wien, 1982. F. Khalatbari, ‘Market imperfections and optimal rate of depletion of an exhaustible resource’, Economica, Vol 44, 1976, pp 409414. William D. Nordhaus, ‘The allocation of energy resources’, Brookings Papers on Economic Activity, Vol 3, 1973. Robert S. Pindyck, ‘Gains to producers from cartelization’, Review of Economics and Statistics, Vol 60, 1978, pp 238-25 1. Robert S. Pindyck, ‘The optimal exploration and production of nonrenewable resources’, Journal of Political Economy, Vol 86, 1978, pp 841-861. Arthur J. Robson, ‘Sequential exploitation of uncertain deposits of a depletable resource’, Journal of Economic Theory, Vol 21, 1979, pp 88-l IO. Walter Rudin, Principles cf Mathematical Analysis, McGraw Hill, 1953. Stephen W. Salant, ‘Staving off the backstop: dynamic limit pricing with a kinked demand curve, in R. S. Pindyck, ed, Adounw.v in the Economics of’ Energy Resources, JAI Press, Vol 2, pp 187-204.
14
given
24%
(Al8)
_:::I
15
16
17
and
Joseph E. Stiglitz and Partha Dasgupta, ‘Market structure and resource extraction under uncertainty’, Scandinavian Journal ojEconomics, Vol83,198 1, pp 3 18-333. Joseph E. Stiglitz and Partha Dasgupta, ‘Market structure and resource depletion: A contribution to the theory of intertemporal monopolistic competition’, Journal of Economic Theory, Vol 28, 1981, pp 128-164. Joseph E. Stiglitz and Partha Dasgupta, ‘Strategic considerations in invention and innovation: the case of natural resources’, Econometrica, Vol 51, 1983, pp 1439-1448. The Review of’ Economic Studies, Symposium on the Economics of Exhaustible Resources, 1974.
ENERGY
ECONOMICS
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1988