- Email: [email protected]
Optimal sequencing of resource pools under uncertainty
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
17,83-92
(1989)
Optimal Sequencing of Resource Pools under Uncertainty’ TROND E. OLSEN Department
of Economics,
Universiv
of Bergen,
N-501 4 Bergen,
Norway
AND Gumm Chr.
Michelsen
Institute,
Centre for Petroleum Bergen,
S~N~LAND Economics, Norway
Fantoftveien
38, N-5036
Fantoft,
Received October 6,1987; revised April 22,1988 This paper characterizes the optimal scheduling of extraction from two reserves under price and production uncertainty. An essential step in the analysis is the characterization of the optimal “switching date” and the associated “flexibility value”; i.e., the value of the option to switch production from the first to the second reserve. 0 1989 Academic PW., IIIC.
1. INTRODUCTION This study is motivated by a concrete problem of oil field development encountered on the Norwegian Continental Shelf. The field consists of two closely situated reserves. The operator is considering two main development programs: (1) draining both reserves from one fixed production platform; or (2) draining the reserves sequentially from a mobile platform. The latter alternative seems attractive because of its inherent flexibility. In order to compare the two alternatives, the value of this flexibility must be assessed. This is the problem that we address. The salient features of the problem are as follows: Two reserves are available at different locations and must be exploited sequentially. Specifically, there is available only one platform, which at some point in time must be moved from one reserve to the other. We assume that the move is irreversible; i.e., only one change of location is feasible. The first problem is to characterize the optimal “stopping rule,” i.e., the rule which specifies when the move is to take place. If this rule is found, one can then (at least in principle) calculate the value of the optimal program, for each of the two possible sequential orderings of the two reserves. For each of the orderings, the value of flexibility can thus be assessed. Finally, the optimal ordering can be determined by comparing the two values. In the model and analysis presented here, we have sacrificed much realism in order to obtain analytically tractable solutions. This approach is chosen because we want to illustrate the value of flexibility as explicitly as possible. Specifically, the following assumptions are invoked: (Al) A change of platform location is irreversible. (A2) The resource price follows a geometric diffusion. ‘lbis paper has benefited considerably from the comments of two anonymous referees. We are responsible for any errors that may remain. 83
0095~06%/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
84
OLSEN AND STENSLAND
(A3) The production profiles are exogenous and follow geometric diffusions. (A4) There are neither costs associated with production nor with changing platform location. Clearly, the assumption (A.4) is not realistic. As a matter of fact, in our motivating concrete development problem, the costs of changing platform location are substantial, and it is precisely those costs which make (Al) a realistic assumption. In this theoretical analysis we ignore relocation costs because that allows us to obtain explicit solutions. However, we are fairly confident that these solutions illustrate general principles, which continue to be valid in more complex and realistic models. To our best knowledge, the problem presented here has not been analyzed in the resource economics literature. A somewhat related study is that of Robson [7], which analyzes socially optimal extraction programs when different pools are of uncertain size. This analysis is not directly relevant for the present problem for two reasons: (a) the decision maker under consideration is a price taker; and (b) the platform technology puts additional restrictions on the extraction policy (e.g., strictly sequential extraction and no repeated switchings between pools). Since the problem at hand involves choosing an optimal date to relocate the platform, the literature on “optimal stopping” clearly becomes relevant. We rely heavily on this literature, and in particular on a recent contribution by McDonald and Siegel [4]. The paper is organized as follows. The model is presented in Section 2. Applying a result due to McDonald and Siegel [4], the optimal stopping rule is characterized in Section 3. It is seen that the optimal rule has a quite simple and economically appealing form. The optimal sequential ordering is analyzed in Section 4. An interesting feature is that, ceteris parabus, the reserve having the largest variance (of the instantaneous rate of change of production) should be given first priority. Section 5 contains a brief discussion of optimal policies when repeated switchings between reserves are feasible. 2. THE MODEL
Production Let q: denote production (= rate of extraction) from reserve i, i = 1,2. It is assumed that, once production from reserve i is started, the extraction path follows a geometric diffusion process with negative drift: dq;/q; = - ai dt + ai dB;,
aj, q > 0, i = 1,2.
(2.1)
Here B: is Brownian motion. The production path is by assumption exogenous, and can only be modified by completely shutting off the well. It is also assumed that this action is irreversible. Equation (2.1) can be solved (see, e.g., Oksendal [lo]) to yield the following explicit formula for the process 41: qf = qiexp(( -6; - af/2)t
+ ajB:),
B; = 0.
(2.2)
SEQUENCING
OF
RESOURCE
85
POOLS
Having observed production for s Q t, expected future production is given by E( qf+,(qf) = qf . exp( -Q).
The expected production
rate thus exhibits geometric decline.
Price
The resource price path is also assumed to follow a geometric diffusion process*: dp,/p, = /Adt + y dB;.
(2.3)
The interest rate (r), which is assumed constant, represents the rate used by a well-diversified decision maker. Typically (e.g., in the Capital Asset Pricing Model), r will depend on the risk free interest rate and the covariance between the market portfolio and the oil price. Production uncertainty is assumed independent of the market portfolio and is therefore irrelevant. The present value price p,eArt then follows a geometric diffusion with drift coefficient p - r. In particular, the expected present value price satisfies Eop,e-” = p. exp((p - r)t). It will be assumed that p < r, such that the expected present value price declines over time. Moreover, it is assumed that pt, q:, and q,? are stochastitally independent. It would probably not be technically very difficult to introduce covariance between the processes. However, for the problem at hand the independence assumption seems quite reasonable, since this is a problem where production uncertainty is revealed only on the project which is active. The Value of a Reserve
Given the assumptions above, we can calculate the value of “remaining tion” from each reserve. Assuming p - r - Si < 0, the formula is E
mqjpse-r(s-t)
dslqf, pt
= ptqf/( 6i + r - /J).
produc-
(2.4)
The formula is shown by noting that, for given qt, pt > 0, the variable X, = is a geometric diffusion with drift coefficient p - r - 6. Iqt+sPt+se -“/(qrpt)] Hence, its expected value is E,X, = exp((p - r - 6)s). For p- r - 6 < 0 we have jFE,,X, dr = l/(S + r - I), hence (2.4) follows. It should be noted that the value in (2.4) is the value of future production, given that the well is never shut off. Define F, to be the conditional expected value of “remaining production” at reserve 1, given complete information about price and production to date t. From *Geometric and by Pindyck
price [6].
processes
for natural
resources
have been used,
e.g., by Brennan
and Schwartz
[l]
86
OLSEN
(2.1)-(2.4)
AND
STENSLAND
it follows that this value also is a geometric diffusion:
dF,/F, = (/A - 6,) dt + TdBI,
7 = (0; + y2y2,
B, = ( olB; + yB;)/c (2.5)
Define V, to be the expected value of the non-producing reserve at date t, conditional on the prevailing price pt. From (2.4) we then obtain 4P,
= Pax4
+ r - p),
ij; = E(q;).
(2.6)
Notice that the value of the non-producing reserve depends on the prior distribution of the initial production rate qi only through the mean rate $. Since the value V, thus is proportional to the current price pI, it immediately follows that y satisfies the diffusion equation dy/K
= p dt + ydB,!
(2.7)
Note that the two values F, and V, are correlated, since they both depend on the common price p,. The instantaneous correlation between their respective rates of change-i.e., between 7dB, and y dB:---is p = y/T = y/( 01’ + y2y2.
3. THE OPTIMAL
STOPPING
(2.8)
RULE
In this section we consider the problem of finding an optimal rule for the operator’s decision to switch from the first to the second reserve. It should be emphasized that the sequential ordering of the reserves is assumed to be given exogenously. (The optimal choice of ordering is considered in Section 4.) Thus we assume that for every date t > 0, the instantaneous production rate for reserve 1 (4:) is observable, and hence known at date t. Before switching, nothing is learned about production rates for reserve 2. In particular, the initial production rate for reserve 2 (4:) is uncertain. From the definitions of the values F, and K in Section 2 it now follows that, if the operator switches at date T, he obtains a total expected payoff F~ + E,,[ eCrT( V, - F,)] .
(3.1)
Recall that K is the expected value of reserve 2, given that production starts at date t, when the price is pt. If production is switched from reserve 1 to reserve 2 at date t, the associated gross gain is captured by K, while the costs are captured by 4, the profits foregone by (irreversibly) shutting down reserve 1. Formula (3.1) emphasizes that the operator should choose his decision rule so as to maximize the expected value of the net gain E, ( evrT( V, - F,)). It should be noted here that if one followed the “naive” rule of switching as soon as the gross gain exceeds costs, i.e., as soon as V, z F,, the expected value of the net
SEQUENCING
OF
RESOURCE
87
POOLS
gain would be zero. This simple decision rule thus makes the switching option worthless. Intuition then suggests that in order to obtain a positive value, switching should only occur if the gross gain V, exceeds costs Ft by a positive amount. Moreover, due to the constant interest rate, the maximal value should be obtained by basing the switching rule on the observed difference V, - F,. Under the given assumptions, this intuition can indeed be verified. The problem of choosing the optimal stopping rule for (3.1) when both V, and 4 follow geometric diffusions has been solved in the literature (McDonald and Siegel [4]). For p < r, the optimal rule is to switch the first time the ratio r/;/Ft reaches a barrier C, where C is determined by the parameters of the two processes [4, p. 7131. Using (2.9, (2.7), and (2.8) it follows that the barrier C in this application is given by c = &/(& - l), E = [( 6,/d - i)’ + 2(r + S, - j~)/o*]~‘* CT*= y* + r* - 2py~ = at.
+ i - S,/a*,
(3.2)
Note that for /J < r we have C > 1, hence the optimal rule requires that one should switch only if gross benefits V, (the value of reserve 2) strictly exceeds costs F, (the value of remaining production from reserve 1). More precisely, switching should take place only if the difference between benefits and costs exceeds some (constant) multiple of costs; i.e., < - F, > kF,, K = C - 1 > 0. Whenever the benefit-cost difference is below this multiple of costs, the value of the switching option is larger than the net benefits obtained by immediate switching. Given the optimal stopping rule, one can calculate explicitly the value of the optimal program (McDonald and Siegel, [4, p. 7131). From (3.1) we obtain the optimal value
8, + (C - l)J,[(WG)/C]e~
(K,/F, 6 C).
(3.3)
The last term in (3.3) reflects the value of the switching option. As noted by McDonald and Siegel [4], both the value of the switching option and the critical level of V/F at which switching should occur, are increasing functions of the parameter a*-the instantaneous variance of V/F. In our application, this parameter is given by q,2 the variance of the instantaneous rate of change for production from reserve 1. From (2.4), (2.9, and (2.6) it is seen that in our application, the ratio V,/F, is given by F/l;; = (&/q:)(WA22),
A, = ai + r - p.
(3.4)
The assumption p < r implies that the expected present value price E,p, exp( - rt) is decreasing. Given this assumption, we see that the optimal rule is to switch the first time the observed production rate from reserve 1 satisfies
where C is given by (3.2).
OLSEN AND STENSLAND
88
Recall that q: is assumed to be observable at date t, t 2 0. In particular, q; is assumed to be known at date t = 0. The preceding analysis is thus best interpreted as a characterization of the ex post optimal switching rule, given that the sequential ordering of the two reserves has been chosen. From (3.3)-(3.5) we see that the value of the ex post optimal program is w=
(p,q~/A,)(l+(C-l)[K/q~l’)
ifq+K
if q: < K.
i P&?~2
(3.6)
Note that if the initial production rate from reserve 1 (46) is revealed to be below the critical rate K given in (3.5), it is optimal to shut off the first well immediately, and switch production to reserve 2. The value of this program is given by the expected present value of extraction from reserve 2, and this is captured in the second line of (3.6). If the initial production rate qh exceeds the critical rate K, it is optimal to postpone switching. Note that without the switching option, the (expected) value of reserve 1 is p,,qA/A1. Hence the value of the switching option is captured by the second term in the first line of (3.6). Before proceeding to the optimal sequencing problem, we add here some remarks on comparative statics results. It was noted by McDonald and Siegel [4, p. 7141 that the value of the switching option (given here by the second term in the first line of (3.6)) is a decreasing function of the drift coefficient for the process F,. In our application this coefficient is p - a,, where p is the price drift coefficient, and 6, is the decline rate for reserve 1. The larger is a,, the smaller is obviously the drift coefficient for F,, and hence the larger is the value of the switching option. However, it can be seen that the total value W of the combined extraction and switching program is a decreasing function of 6,.
This can be seen as follows. For a given decline rate 6, it is optimal to switch the first time the associated extraction rate satisfies q1 < K. Denote this stopping time by T. Let 6f=S,--6, 6>0, and let q’,” be the (diffusion) extraction rate associated with this smaller decline rate. Since ql, ’ = q1 . exp( at), the time T is also a stopping time for q’*” -but not the optimal stopping time, of course. Suppose we stop at time T. At every instant before T, the process q’,” yields a higher revenue than the process q ‘. Since the revenue after time T (the value of reserve 2) is independent of a,, it follows that the total value W” associated with the smaller decline rate 8; is larger than the total value W associated with 6,, (6, > 8;). By a similar argument it is easily seen that the total value W is also a decreasing function of the decline rate (8,) for the second reserve. The only variance term entering the value function W is a:-the variance of (the instantaneous rate of change of) production from reserve 1. As we noted above it follows from McDonald and Siegel that the part of W which accounts for the value of the switching option is an increasing function of this parameter. Taking account of the fact that the barrier K is a decreasing function of a;, it then follows that W itself is an increasing function of ~1”. More precisely we have 6: > u1” *
and the inequality
w( 4;; 6:) 2 w( 4;; a?),
is strict for qh > K(6;).
all q: > 0,
(3.7)
SEQUENCING
OF RESOURCE POOLS
89
It may at first sight be surprising that the variance of the initial production rate of reserve 2 does not influence the option value. However, since no decisions depend on the realization of this initial production rate, potential variations in this rate are of no value. (4:)
4. OPTIMAL
SEQUENCING
In the previous sections we have analyzed the optimal switching problem, given that a particular ordering of the reserves has been chosen beforehand. Now consider the optimal ordering problem: Which site (reserve) should be chosen as the first location for the production platform? The ex ante expected value of the ordering (1,2) is E W, where W is the ex post expected value given by (3.6). The ex ante expectation EW is with respect to the distribution of the initial production rate (4;) from reserve 1. Let Z be defined as the ex post expected value of the reverse ordering (2,l). Z is given by a formula completely analogous to (3.6), with suitable modifications in the definitions of the parameters E, C, and K. The ex ante expected value of the ordering (2,l) is then EZ, where the expectation is with respect to the initial production rate (4:) from reserve 2. The optimal ordering is obtained by comparing the two ex ante values EW and EZ. The optimal ordering is determined by the probability distributions of the initial production rates (qh), and the model parameters r, ~1, Si, ei. A complete classification is tedious and will not be given here. However, we wish to point out two interesting results. First, if the two reserves are identical in all respects, except for the variances (crf ) of the respective rates of change of production, the reserue having the largest variance should be given first priority.
This result is a simple consequence of the comparative statics results that the ex post value W is an increasing function of the variance c$-see (3.7)-and is independent of the variance 022.For suppose crf > u.j. Then for every realization of the initial production rate q,, for the reserve taken first, the ex post values W and Z associated with the orderings (1,2) and (2,l) respectively, satisfy W a Z, with strict inequality for q,, > K(uf). Hence the ex ante values satisfy E W >, EZ, with strict inequality if q,, > K(uf) with positive probability. This proves the result that the reserve having the largest variance for the instantaneous rate of change of production should be given first priority. Next suppose the reserves are identical, except that reserve 1 has a more risky distribution (in the Rothschild-Stiglitz sense) for the initial rate of production. Then the reserve having the more risky distribution should be given first priority. This result follows immediately from noting that the ex post value function W in (3.6) is a convex function of the realized initial production rate qO for the reserve taken first. Giving priority to the more risky reserve therefore increases the ex ante value EW. Results of this type are quite common in the literature on optimal search-see, e.g., Weitzman [8]-and is perhaps best interpreted as preferences for larger variety. However, it is of course not always the case that more risk is preferred to less risk. To give a counterexample in a resource context we may note that Flam and Olsen [2] showed that-in a model where extraction from each pool is constant (e.g., due to capacity constraints)-increased risk in the stock size is disadvantageous.
90
OLSEN AND STENSLAND
5. RESWITCHING
It has so far been assumed that the change of production location is an irreversible act. Although this is probably a realistic description for the concrete development program which motivated this study, it is certainly interesting to examine the consequences of relaxing this assumption. So assume now there are no restrictions on the number of times we may switch production back and forth between the two reserves. Brennan and Schwartz [l] consider the problem of a perpetual switching option, in which a mine is opened or closed depending upon the relationship between price and extraction costs. Except for the case where the resource is infinite, they rely on numerical solution methods. Elsewhere [S], we have pointed out the analytic solution to the problem of irreversibly stopping production when both price and production follow (geometric) diffusions. Even though there are obvious similarities between the “on-off’ switching problem and the problem of switching back and forth between two income generating activities, the latter turns out to be considerably more complicated than the former, even in the absence of switching costs. At a first glance intuition seems to suggest that in the case of no switching costs, it would be optimal to produce at the reserve having the largest instantaneous rate of production. This policy is in fact no1 optimal. To see why this is so, suppose production from the two reserves have quite different variances. Then, even if current production rates were equal, it would pay to produce from the high-variance reserve, because to do so would increase the probability of realizing higher production rates in the near future. (Low future production rates can be eliminated by switching.) The intuitive argument above is standard in the literature on “bandit-problems” -see, e.g., [9]. Some results from that literature can also be brought to bear on our problem. However, to formulate our switching problem as a bandit problem, it must be assumed that the state of a project is unchanged as long as that project is inactive (e.g., as long as the reserve in question is closed). Thus we must assume that price is deterministic and stationary. Given this assumption, we may normalize the price to be pt = 1, all t. Consider now the following thought experiment for reserve i, i = 1,2. Suppose we are given the option to terminate production from reserve i and receive a fixed payment mj. The optimal time to terminate would then be the stopping time T which maximizes
/oTq:ePrtdt + eerTmi
= Fi + EO[ePrT(m, - Fi)],
(5.1)
where F,’ = q:/(Si + r). As pointed out in Section 3, it is optimal to stop the first time mi/F,,’ 2 Ci, where the barriers Cj, i = 1,2 are defined by equations analogous to (3.2). Note that C, > C, if uf > u; and the reserves are otherwise identical. The optimal value of (5.1) is
Fl + (Ci - l)Fi[(mJFd)/Ci]‘,
(m/F:
< Ci),
i = 1,2.
(5.2)
SEQUENCING
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POOLS
91
Now suppose the termination reward mi is chosen to be the smallest reward for which the decision maker is indifferent between terminating and continuing (optimally, of course) the project. This critical reward is the Gittins-index for project (reserve) i, and it is easily seen, e.g., from (5.2), that the index in this case is given by mi = Ci . F; = Ci - q;/(Si
+ r),
i = 1,2,
(5.3)
where the second equality follows from the definition of Fi above. The usefulness of these indices follows from the observation that in the setting described here, the optimal policy is to activate at each date the reserve having the largest Gittins-index (see, e.g., Karatzas [3]). Note that the indices depend on the “states,” i.e., current production rates, of the respective reserves. Assume now that the reserves are identical, except that production from reserve 1 has the larger variance (a: > u,‘). Since then C, > C,, it follows that reserve 2 should be activated only if q2/q1 > C,/C, > 1. This result confirms the intuitive argument given previously in this section. 6. CONCLUDING
REMARKS
This paper has characterized-at a high level of abstraction-the optimal scheduling of extraction from two reserves under price and production uncertainty -given that the reserves are to be exploited sequentially. (Sequentiality is a natural requirement in the problem motivating this study.) An essential step in the analysis was the characterization-under some strong assumptions-of the optimal “switching date” and the associated “flexibility value”; i.e., the value of the option to switch production from the first to the second reserve. This option value is often overlooked in resource management practice. This paper may perhaps contribute to a better understanding of option values in resource management. On the theoretical level, there are two contributions in this paper: (i) an explicit model of price and production stochastics, which allows the option value to be characterized in terms of parameters of the price and production processes; and (ii) the characterization of the optimal sequence (Section 4). However, it is not satisfactory that this was accomplished only by invoking the assumption (A4) that switching costs (associated with relocating the platform) were nil. The generalization of these results to the case of positive switching costs is a challenging task for future research.
REFERENCES 1. M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, J. Business 58,135-157 (1985). 2. S. D. F&m and T. E. Olsen, Scheduling and taxation of resource deposits, The Energy J. 6 (Special Tax Issue), 137-143 (1985). 3. I. Karatzas, Gittins indices in the dynamic allocation problem for diffusion processes, Ann. Probub. 12, No. 1, 173-192 (1984). 4. R. McDonald and D. Siegel, The value of waiting to invest, Quart. J. Econom. 101, November, 707-727 (1986). 5. T. E. Olsen and G. Stensland, Optimal shut down decision in resource extraction, Econom. Lett. 26, 215-218 (1988).
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6. R. S. Pindyck, The optimal production of an exhaustible resource when price is exogenous and stochastic, Scund. J. Econom. 83, 277-288 (1981). 7. A. Robson, Sequential exploitation of uncertain deposits of a depletable natural resource, J. Econom. Theory 21, 88-110 (1979). 8. M. L. Weitzman, Optimal search for the best alternative, Econometrica 47, No. 3, 641-655 (1979). 9. P. Whittle, “Optimization over Time,” Vol. 1, Wiley, New York (1982). 10. B. Oksendal, “Stochastic Differential Equations. An Introduction with Applications,” SpringerVerlag, Berlin (1985).
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
17,83-92
(1989)
Optimal Sequencing of Resource Pools under Uncertainty’ TROND E. OLSEN Department
of Economics,
Universiv
of Bergen,
N-501 4 Bergen,
Norway
AND Gumm Chr.
Michelsen
Institute,
Centre for Petroleum Bergen,
S~N~LAND Economics, Norway
Fantoftveien
38, N-5036
Fantoft,
Received October 6,1987; revised April 22,1988 This paper characterizes the optimal scheduling of extraction from two reserves under price and production uncertainty. An essential step in the analysis is the characterization of the optimal “switching date” and the associated “flexibility value”; i.e., the value of the option to switch production from the first to the second reserve. 0 1989 Academic PW., IIIC.
1. INTRODUCTION This study is motivated by a concrete problem of oil field development encountered on the Norwegian Continental Shelf. The field consists of two closely situated reserves. The operator is considering two main development programs: (1) draining both reserves from one fixed production platform; or (2) draining the reserves sequentially from a mobile platform. The latter alternative seems attractive because of its inherent flexibility. In order to compare the two alternatives, the value of this flexibility must be assessed. This is the problem that we address. The salient features of the problem are as follows: Two reserves are available at different locations and must be exploited sequentially. Specifically, there is available only one platform, which at some point in time must be moved from one reserve to the other. We assume that the move is irreversible; i.e., only one change of location is feasible. The first problem is to characterize the optimal “stopping rule,” i.e., the rule which specifies when the move is to take place. If this rule is found, one can then (at least in principle) calculate the value of the optimal program, for each of the two possible sequential orderings of the two reserves. For each of the orderings, the value of flexibility can thus be assessed. Finally, the optimal ordering can be determined by comparing the two values. In the model and analysis presented here, we have sacrificed much realism in order to obtain analytically tractable solutions. This approach is chosen because we want to illustrate the value of flexibility as explicitly as possible. Specifically, the following assumptions are invoked: (Al) A change of platform location is irreversible. (A2) The resource price follows a geometric diffusion. ‘lbis paper has benefited considerably from the comments of two anonymous referees. We are responsible for any errors that may remain. 83
0095~06%/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
84
OLSEN AND STENSLAND
(A3) The production profiles are exogenous and follow geometric diffusions. (A4) There are neither costs associated with production nor with changing platform location. Clearly, the assumption (A.4) is not realistic. As a matter of fact, in our motivating concrete development problem, the costs of changing platform location are substantial, and it is precisely those costs which make (Al) a realistic assumption. In this theoretical analysis we ignore relocation costs because that allows us to obtain explicit solutions. However, we are fairly confident that these solutions illustrate general principles, which continue to be valid in more complex and realistic models. To our best knowledge, the problem presented here has not been analyzed in the resource economics literature. A somewhat related study is that of Robson [7], which analyzes socially optimal extraction programs when different pools are of uncertain size. This analysis is not directly relevant for the present problem for two reasons: (a) the decision maker under consideration is a price taker; and (b) the platform technology puts additional restrictions on the extraction policy (e.g., strictly sequential extraction and no repeated switchings between pools). Since the problem at hand involves choosing an optimal date to relocate the platform, the literature on “optimal stopping” clearly becomes relevant. We rely heavily on this literature, and in particular on a recent contribution by McDonald and Siegel [4]. The paper is organized as follows. The model is presented in Section 2. Applying a result due to McDonald and Siegel [4], the optimal stopping rule is characterized in Section 3. It is seen that the optimal rule has a quite simple and economically appealing form. The optimal sequential ordering is analyzed in Section 4. An interesting feature is that, ceteris parabus, the reserve having the largest variance (of the instantaneous rate of change of production) should be given first priority. Section 5 contains a brief discussion of optimal policies when repeated switchings between reserves are feasible. 2. THE MODEL
Production Let q: denote production (= rate of extraction) from reserve i, i = 1,2. It is assumed that, once production from reserve i is started, the extraction path follows a geometric diffusion process with negative drift: dq;/q; = - ai dt + ai dB;,
aj, q > 0, i = 1,2.
(2.1)
Here B: is Brownian motion. The production path is by assumption exogenous, and can only be modified by completely shutting off the well. It is also assumed that this action is irreversible. Equation (2.1) can be solved (see, e.g., Oksendal [lo]) to yield the following explicit formula for the process 41: qf = qiexp(( -6; - af/2)t
+ ajB:),
B; = 0.
(2.2)
SEQUENCING
OF
RESOURCE
85
POOLS
Having observed production for s Q t, expected future production is given by E( qf+,(qf) = qf . exp( -Q).
The expected production
rate thus exhibits geometric decline.
Price
The resource price path is also assumed to follow a geometric diffusion process*: dp,/p, = /Adt + y dB;.
(2.3)
The interest rate (r), which is assumed constant, represents the rate used by a well-diversified decision maker. Typically (e.g., in the Capital Asset Pricing Model), r will depend on the risk free interest rate and the covariance between the market portfolio and the oil price. Production uncertainty is assumed independent of the market portfolio and is therefore irrelevant. The present value price p,eArt then follows a geometric diffusion with drift coefficient p - r. In particular, the expected present value price satisfies Eop,e-” = p. exp((p - r)t). It will be assumed that p < r, such that the expected present value price declines over time. Moreover, it is assumed that pt, q:, and q,? are stochastitally independent. It would probably not be technically very difficult to introduce covariance between the processes. However, for the problem at hand the independence assumption seems quite reasonable, since this is a problem where production uncertainty is revealed only on the project which is active. The Value of a Reserve
Given the assumptions above, we can calculate the value of “remaining tion” from each reserve. Assuming p - r - Si < 0, the formula is E
mqjpse-r(s-t)
dslqf, pt
= ptqf/( 6i + r - /J).
produc-
(2.4)
The formula is shown by noting that, for given qt, pt > 0, the variable X, = is a geometric diffusion with drift coefficient p - r - 6. Iqt+sPt+se -“/(qrpt)] Hence, its expected value is E,X, = exp((p - r - 6)s). For p- r - 6 < 0 we have jFE,,X, dr = l/(S + r - I), hence (2.4) follows. It should be noted that the value in (2.4) is the value of future production, given that the well is never shut off. Define F, to be the conditional expected value of “remaining production” at reserve 1, given complete information about price and production to date t. From *Geometric and by Pindyck
price [6].
processes
for natural
resources
have been used,
e.g., by Brennan
and Schwartz
[l]
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OLSEN
(2.1)-(2.4)
AND
STENSLAND
it follows that this value also is a geometric diffusion:
dF,/F, = (/A - 6,) dt + TdBI,
7 = (0; + y2y2,
B, = ( olB; + yB;)/c (2.5)
Define V, to be the expected value of the non-producing reserve at date t, conditional on the prevailing price pt. From (2.4) we then obtain 4P,
= Pax4
+ r - p),
ij; = E(q;).
(2.6)
Notice that the value of the non-producing reserve depends on the prior distribution of the initial production rate qi only through the mean rate $. Since the value V, thus is proportional to the current price pI, it immediately follows that y satisfies the diffusion equation dy/K
= p dt + ydB,!
(2.7)
Note that the two values F, and V, are correlated, since they both depend on the common price p,. The instantaneous correlation between their respective rates of change-i.e., between 7dB, and y dB:---is p = y/T = y/( 01’ + y2y2.
3. THE OPTIMAL
STOPPING
(2.8)
RULE
In this section we consider the problem of finding an optimal rule for the operator’s decision to switch from the first to the second reserve. It should be emphasized that the sequential ordering of the reserves is assumed to be given exogenously. (The optimal choice of ordering is considered in Section 4.) Thus we assume that for every date t > 0, the instantaneous production rate for reserve 1 (4:) is observable, and hence known at date t. Before switching, nothing is learned about production rates for reserve 2. In particular, the initial production rate for reserve 2 (4:) is uncertain. From the definitions of the values F, and K in Section 2 it now follows that, if the operator switches at date T, he obtains a total expected payoff F~ + E,,[ eCrT( V, - F,)] .
(3.1)
Recall that K is the expected value of reserve 2, given that production starts at date t, when the price is pt. If production is switched from reserve 1 to reserve 2 at date t, the associated gross gain is captured by K, while the costs are captured by 4, the profits foregone by (irreversibly) shutting down reserve 1. Formula (3.1) emphasizes that the operator should choose his decision rule so as to maximize the expected value of the net gain E, ( evrT( V, - F,)). It should be noted here that if one followed the “naive” rule of switching as soon as the gross gain exceeds costs, i.e., as soon as V, z F,, the expected value of the net
SEQUENCING
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gain would be zero. This simple decision rule thus makes the switching option worthless. Intuition then suggests that in order to obtain a positive value, switching should only occur if the gross gain V, exceeds costs Ft by a positive amount. Moreover, due to the constant interest rate, the maximal value should be obtained by basing the switching rule on the observed difference V, - F,. Under the given assumptions, this intuition can indeed be verified. The problem of choosing the optimal stopping rule for (3.1) when both V, and 4 follow geometric diffusions has been solved in the literature (McDonald and Siegel [4]). For p < r, the optimal rule is to switch the first time the ratio r/;/Ft reaches a barrier C, where C is determined by the parameters of the two processes [4, p. 7131. Using (2.9, (2.7), and (2.8) it follows that the barrier C in this application is given by c = &/(& - l), E = [( 6,/d - i)’ + 2(r + S, - j~)/o*]~‘* CT*= y* + r* - 2py~ = at.
+ i - S,/a*,
(3.2)
Note that for /J < r we have C > 1, hence the optimal rule requires that one should switch only if gross benefits V, (the value of reserve 2) strictly exceeds costs F, (the value of remaining production from reserve 1). More precisely, switching should take place only if the difference between benefits and costs exceeds some (constant) multiple of costs; i.e., < - F, > kF,, K = C - 1 > 0. Whenever the benefit-cost difference is below this multiple of costs, the value of the switching option is larger than the net benefits obtained by immediate switching. Given the optimal stopping rule, one can calculate explicitly the value of the optimal program (McDonald and Siegel, [4, p. 7131). From (3.1) we obtain the optimal value
8, + (C - l)J,[(WG)/C]e~
(K,/F, 6 C).
(3.3)
The last term in (3.3) reflects the value of the switching option. As noted by McDonald and Siegel [4], both the value of the switching option and the critical level of V/F at which switching should occur, are increasing functions of the parameter a*-the instantaneous variance of V/F. In our application, this parameter is given by q,2 the variance of the instantaneous rate of change for production from reserve 1. From (2.4), (2.9, and (2.6) it is seen that in our application, the ratio V,/F, is given by F/l;; = (&/q:)(WA22),
A, = ai + r - p.
(3.4)
The assumption p < r implies that the expected present value price E,p, exp( - rt) is decreasing. Given this assumption, we see that the optimal rule is to switch the first time the observed production rate from reserve 1 satisfies
where C is given by (3.2).
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88
Recall that q: is assumed to be observable at date t, t 2 0. In particular, q; is assumed to be known at date t = 0. The preceding analysis is thus best interpreted as a characterization of the ex post optimal switching rule, given that the sequential ordering of the two reserves has been chosen. From (3.3)-(3.5) we see that the value of the ex post optimal program is w=
(p,q~/A,)(l+(C-l)[K/q~l’)
ifq+K
if q: < K.
i P&?~2
(3.6)
Note that if the initial production rate from reserve 1 (46) is revealed to be below the critical rate K given in (3.5), it is optimal to shut off the first well immediately, and switch production to reserve 2. The value of this program is given by the expected present value of extraction from reserve 2, and this is captured in the second line of (3.6). If the initial production rate qh exceeds the critical rate K, it is optimal to postpone switching. Note that without the switching option, the (expected) value of reserve 1 is p,,qA/A1. Hence the value of the switching option is captured by the second term in the first line of (3.6). Before proceeding to the optimal sequencing problem, we add here some remarks on comparative statics results. It was noted by McDonald and Siegel [4, p. 7141 that the value of the switching option (given here by the second term in the first line of (3.6)) is a decreasing function of the drift coefficient for the process F,. In our application this coefficient is p - a,, where p is the price drift coefficient, and 6, is the decline rate for reserve 1. The larger is a,, the smaller is obviously the drift coefficient for F,, and hence the larger is the value of the switching option. However, it can be seen that the total value W of the combined extraction and switching program is a decreasing function of 6,.
This can be seen as follows. For a given decline rate 6, it is optimal to switch the first time the associated extraction rate satisfies q1 < K. Denote this stopping time by T. Let 6f=S,--6, 6>0, and let q’,” be the (diffusion) extraction rate associated with this smaller decline rate. Since ql, ’ = q1 . exp( at), the time T is also a stopping time for q’*” -but not the optimal stopping time, of course. Suppose we stop at time T. At every instant before T, the process q’,” yields a higher revenue than the process q ‘. Since the revenue after time T (the value of reserve 2) is independent of a,, it follows that the total value W” associated with the smaller decline rate 8; is larger than the total value W associated with 6,, (6, > 8;). By a similar argument it is easily seen that the total value W is also a decreasing function of the decline rate (8,) for the second reserve. The only variance term entering the value function W is a:-the variance of (the instantaneous rate of change of) production from reserve 1. As we noted above it follows from McDonald and Siegel that the part of W which accounts for the value of the switching option is an increasing function of this parameter. Taking account of the fact that the barrier K is a decreasing function of a;, it then follows that W itself is an increasing function of ~1”. More precisely we have 6: > u1” *
and the inequality
w( 4;; 6:) 2 w( 4;; a?),
is strict for qh > K(6;).
all q: > 0,
(3.7)
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OF RESOURCE POOLS
89
It may at first sight be surprising that the variance of the initial production rate of reserve 2 does not influence the option value. However, since no decisions depend on the realization of this initial production rate, potential variations in this rate are of no value. (4:)
4. OPTIMAL
SEQUENCING
In the previous sections we have analyzed the optimal switching problem, given that a particular ordering of the reserves has been chosen beforehand. Now consider the optimal ordering problem: Which site (reserve) should be chosen as the first location for the production platform? The ex ante expected value of the ordering (1,2) is E W, where W is the ex post expected value given by (3.6). The ex ante expectation EW is with respect to the distribution of the initial production rate (4;) from reserve 1. Let Z be defined as the ex post expected value of the reverse ordering (2,l). Z is given by a formula completely analogous to (3.6), with suitable modifications in the definitions of the parameters E, C, and K. The ex ante expected value of the ordering (2,l) is then EZ, where the expectation is with respect to the initial production rate (4:) from reserve 2. The optimal ordering is obtained by comparing the two ex ante values EW and EZ. The optimal ordering is determined by the probability distributions of the initial production rates (qh), and the model parameters r, ~1, Si, ei. A complete classification is tedious and will not be given here. However, we wish to point out two interesting results. First, if the two reserves are identical in all respects, except for the variances (crf ) of the respective rates of change of production, the reserue having the largest variance should be given first priority.
This result is a simple consequence of the comparative statics results that the ex post value W is an increasing function of the variance c$-see (3.7)-and is independent of the variance 022.For suppose crf > u.j. Then for every realization of the initial production rate q,, for the reserve taken first, the ex post values W and Z associated with the orderings (1,2) and (2,l) respectively, satisfy W a Z, with strict inequality for q,, > K(uf). Hence the ex ante values satisfy E W >, EZ, with strict inequality if q,, > K(uf) with positive probability. This proves the result that the reserve having the largest variance for the instantaneous rate of change of production should be given first priority. Next suppose the reserves are identical, except that reserve 1 has a more risky distribution (in the Rothschild-Stiglitz sense) for the initial rate of production. Then the reserve having the more risky distribution should be given first priority. This result follows immediately from noting that the ex post value function W in (3.6) is a convex function of the realized initial production rate qO for the reserve taken first. Giving priority to the more risky reserve therefore increases the ex ante value EW. Results of this type are quite common in the literature on optimal search-see, e.g., Weitzman [8]-and is perhaps best interpreted as preferences for larger variety. However, it is of course not always the case that more risk is preferred to less risk. To give a counterexample in a resource context we may note that Flam and Olsen [2] showed that-in a model where extraction from each pool is constant (e.g., due to capacity constraints)-increased risk in the stock size is disadvantageous.
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OLSEN AND STENSLAND
5. RESWITCHING
It has so far been assumed that the change of production location is an irreversible act. Although this is probably a realistic description for the concrete development program which motivated this study, it is certainly interesting to examine the consequences of relaxing this assumption. So assume now there are no restrictions on the number of times we may switch production back and forth between the two reserves. Brennan and Schwartz [l] consider the problem of a perpetual switching option, in which a mine is opened or closed depending upon the relationship between price and extraction costs. Except for the case where the resource is infinite, they rely on numerical solution methods. Elsewhere [S], we have pointed out the analytic solution to the problem of irreversibly stopping production when both price and production follow (geometric) diffusions. Even though there are obvious similarities between the “on-off’ switching problem and the problem of switching back and forth between two income generating activities, the latter turns out to be considerably more complicated than the former, even in the absence of switching costs. At a first glance intuition seems to suggest that in the case of no switching costs, it would be optimal to produce at the reserve having the largest instantaneous rate of production. This policy is in fact no1 optimal. To see why this is so, suppose production from the two reserves have quite different variances. Then, even if current production rates were equal, it would pay to produce from the high-variance reserve, because to do so would increase the probability of realizing higher production rates in the near future. (Low future production rates can be eliminated by switching.) The intuitive argument above is standard in the literature on “bandit-problems” -see, e.g., [9]. Some results from that literature can also be brought to bear on our problem. However, to formulate our switching problem as a bandit problem, it must be assumed that the state of a project is unchanged as long as that project is inactive (e.g., as long as the reserve in question is closed). Thus we must assume that price is deterministic and stationary. Given this assumption, we may normalize the price to be pt = 1, all t. Consider now the following thought experiment for reserve i, i = 1,2. Suppose we are given the option to terminate production from reserve i and receive a fixed payment mj. The optimal time to terminate would then be the stopping time T which maximizes
/oTq:ePrtdt + eerTmi
= Fi + EO[ePrT(m, - Fi)],
(5.1)
where F,’ = q:/(Si + r). As pointed out in Section 3, it is optimal to stop the first time mi/F,,’ 2 Ci, where the barriers Cj, i = 1,2 are defined by equations analogous to (3.2). Note that C, > C, if uf > u; and the reserves are otherwise identical. The optimal value of (5.1) is
Fl + (Ci - l)Fi[(mJFd)/Ci]‘,
(m/F:
< Ci),
i = 1,2.
(5.2)
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Now suppose the termination reward mi is chosen to be the smallest reward for which the decision maker is indifferent between terminating and continuing (optimally, of course) the project. This critical reward is the Gittins-index for project (reserve) i, and it is easily seen, e.g., from (5.2), that the index in this case is given by mi = Ci . F; = Ci - q;/(Si
+ r),
i = 1,2,
(5.3)
where the second equality follows from the definition of Fi above. The usefulness of these indices follows from the observation that in the setting described here, the optimal policy is to activate at each date the reserve having the largest Gittins-index (see, e.g., Karatzas [3]). Note that the indices depend on the “states,” i.e., current production rates, of the respective reserves. Assume now that the reserves are identical, except that production from reserve 1 has the larger variance (a: > u,‘). Since then C, > C,, it follows that reserve 2 should be activated only if q2/q1 > C,/C, > 1. This result confirms the intuitive argument given previously in this section. 6. CONCLUDING
REMARKS
This paper has characterized-at a high level of abstraction-the optimal scheduling of extraction from two reserves under price and production uncertainty -given that the reserves are to be exploited sequentially. (Sequentiality is a natural requirement in the problem motivating this study.) An essential step in the analysis was the characterization-under some strong assumptions-of the optimal “switching date” and the associated “flexibility value”; i.e., the value of the option to switch production from the first to the second reserve. This option value is often overlooked in resource management practice. This paper may perhaps contribute to a better understanding of option values in resource management. On the theoretical level, there are two contributions in this paper: (i) an explicit model of price and production stochastics, which allows the option value to be characterized in terms of parameters of the price and production processes; and (ii) the characterization of the optimal sequence (Section 4). However, it is not satisfactory that this was accomplished only by invoking the assumption (A4) that switching costs (associated with relocating the platform) were nil. The generalization of these results to the case of positive switching costs is a challenging task for future research.
REFERENCES 1. M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, J. Business 58,135-157 (1985). 2. S. D. F&m and T. E. Olsen, Scheduling and taxation of resource deposits, The Energy J. 6 (Special Tax Issue), 137-143 (1985). 3. I. Karatzas, Gittins indices in the dynamic allocation problem for diffusion processes, Ann. Probub. 12, No. 1, 173-192 (1984). 4. R. McDonald and D. Siegel, The value of waiting to invest, Quart. J. Econom. 101, November, 707-727 (1986). 5. T. E. Olsen and G. Stensland, Optimal shut down decision in resource extraction, Econom. Lett. 26, 215-218 (1988).
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6. R. S. Pindyck, The optimal production of an exhaustible resource when price is exogenous and stochastic, Scund. J. Econom. 83, 277-288 (1981). 7. A. Robson, Sequential exploitation of uncertain deposits of a depletable natural resource, J. Econom. Theory 21, 88-110 (1979). 8. M. L. Weitzman, Optimal search for the best alternative, Econometrica 47, No. 3, 641-655 (1979). 9. P. Whittle, “Optimization over Time,” Vol. 1, Wiley, New York (1982). 10. B. Oksendal, “Stochastic Differential Equations. An Introduction with Applications,” SpringerVerlag, Berlin (1985).