Real-options aspects of adjacency constraints

Real-options aspects of adjacency constraints

Forest Policy and Economics 6 (2004) 261 – 270 www.elsevier.com/locate/forpol Real-options aspects of adjacency constraints Nikolaj Malchow-Møller a,...

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Forest Policy and Economics 6 (2004) 261 – 270 www.elsevier.com/locate/forpol

Real-options aspects of adjacency constraints Nikolaj Malchow-Møller a,*, Niels Strange b, Bo Jellesmark Thorsen a,c a

Centre for Economic and Business Research (CEBR), Langelinie Alle´ 17, DK-2100 Copenhagen Ø, Denmark b Skov Landskab, Rolighedsvej 23, DK-1958 Frederiksberg, Denmark c Skov Landskab, Hørsholm Kongevej 11, DK-2970 Hørsholm, Denmark

Abstract An important topic in the forest-management literature is the temporal and spatial arrangement of harvests. The logging of larger tracts of forest land is often found to be detrimental to many environmental as well as social values. For these reasons, constraints are often imposed upon harvesting options on adjacent forest stands. As is often the case with management restrictions, they inflict costs upon the forest owners—in terms of revenues foregone and decreased market values of harvesting. These costs and the associated optimal harvesting plans—subject to the restrictions—have so far only been evaluated using deterministic models. In this paper, we explore the effects of uncertainty on the costs of adjacency restrictions in a rather stylised two-stands real-options model. Furthermore, we examine the optimal harvest rules under adjacency restrictions and uncertainty. Traditional studies addressing real options in a forestry context have applied a single real-option approach. Recent developments by Malchow-Møller and Thorsen (2003) extend the single-dimension case to the handling of two exclusive options. This paper further extends the two-options approach to be able to handle the adjacency problem. We find that costs of adjacency constraints tend to increase with uncertainty and that the optimal harvesting strategies become rather complex in terms of the involved stochastic variables. D 2004 Elsevier B.V. All rights reserved. Keywords: Real options; Geometric Brownian motions; Stochastic control; Dynamic programming; Adjacency constraints; Optimal harvesting rules

1. Introduction Public concerns about the impacts of forest operations on landscape aesthetics, wildlife provision and endangered species protection have been increasing. As a consequence, societies are faced with complex ecosystem management problems due to competing and complementary social values, and interactions

* Corresponding author. Tel.: +45-35466536; fax: +4535466201. E-mail addresses: [email protected] (N. Malchow-Møller), [email protected] (N. Strange), [email protected] (B.J. Thorsen). 1389-9341/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.forpol.2004.03.002

between the social values and timber production returns. Traditionally, the ‘where and when’ forest management approach has primarily been based on independent stand-level harvest models of Faustmann (1849) and extensions such as Hartman (1976). Bowes and Krutilla (1985, 1989) extend the Faustmann –Hartman model to include multiplestand, multiple-use forests, placing emphasis on ‘when to harvest’ an even-aged forest stand. Such multiple-stand models may account for the distribution of stand ages across the forest, but do not explicitly account for spatially important effects of landscape pattern or location on ecological functions.

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Swallow and Wear (1993) show that spatial interactions across the landscape may lead to substantial departures from standard solutions to spatial forest management problems. Swallow et al. (1997) extend Swallow and Wear’s model of spatial interactions to optimise the management of the entire forest ecosystem. Specifically, ecological interactions across stands may generate substitution or complementarity effects, which must be balanced in determining the optimal harvest age (Koskela and Ollikainen, 2001; Swallow et al., 1997). In practice, ecological and social goals are pursued by imposing adjacency constraints on clear-cut size, minimum distances between clear cuts, green-up constraints, or exclusion periods, see, e.g., Thompson et al. (1973), Jamnick et al. (1990), Bettinger et al. (1998), Murray (1999) and Crowe et al. (2003). These adjacency restrictions may force the private forest owner into non-optimal harvest timings seen from a more narrow profit-maximising perspective, which ignores the ecological and social gains. Hence, although the restrictions may well be optimal seen from the point of view of society—exactly because they take into account the social and ecological externalities—they typically inflict costs upon the private forest owner, since the associated gains do not accrue (exclusively) to him. Our focus in the following is the private forest owner who does not take into account the social and ecological externalities, but maximises net timber production returns subject to the imposed adjacency constraints. To him, the adjacency constraints are costly in terms of reduced timber production values, and may affect his harvesting decisions. Considering a deterministic case, an imposed harvest exclusionperiod constraint of period length s introduces the economic planning problem of finding the set of rotation ages, which maximises the sum of net present values of the adjacent stands under this constraint. This is also the solution that minimises the costs of the adjacency constraint. These costs and the resulting optimal harvesting plans have so far only been evaluated using deterministic models in the scientific literature. An important contribution that we make in this paper is to explore the effects of uncertainty on the costs of adjacency restrictions and the optimal harvest timing of adjacent stands subject to the restrictions. Because uncertainty is

inherent in real-world decisions, this will provide a better idea of the true costs, which should be balanced against the benefits pursued by imposing the adjacency constraints. The decision to harvest a stand is irreversible – once harvested the stand cannot be restored-where – as the decision to postpone harvesting is not irreversible. With known costs of harvesting and uncertainty about revenues, the decision problem possesses the characteristics of what has been widely known as real-option problems (Dixit and Pindyck, 1994). The real-options framework as applied within natural-resource economics has its origin in the seminal studies by Henry (1974) and Arrow and Fisher (1974), who treated a classical case of land-use decisions under uncertainty. The latter study formulated the idea of a quasi option value related to the option to postpone an irreversible decision in order to resolve at least part of the uncertainty related to the future returns. This quasioption value is now widely known as ‘the value of waiting’ (Dixit and Pindyck, 1994). Within forestry economics, most real-options studies have treated the question of replacing a forest stand with the same or another use. Examples include, among many others, Zinkhan (1991) and Thomson (1992). They both discuss the use of the Black – Scholes formula (Black and Scholes, 1973) for valuing the option to harvest a forest and replace it with another land use, where the value of harvesting the forest is stochastic. Reed (1993) studies the option to replace an old-growth forest with another land use, the value of both being stochastic. In a similar study, Abildtrup and Strange (2000) analyse the decision to harvest a forest stand, which produces externalities of uncertain future values, and replace it with an intensive production, the value of which is also stochastic. Thorsen (1999) analyses the effects of afforestation subsidies by taking into consideration that afforestation may be considered a real option on an uncertain investment. All of these studies address the problem of replacing an existing land use with another in a situation where the value of one (or both) of these is (are) stochastic. With only one stochastic value, the problem becomes very similar—if not identical—to the core models of McDonald and Siegel (1986) and Dixit and Pindyck (1994). If both values are stochastic, a homogeneity property of the value function can

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typically be invoked. This property allows for a reformulation of the problem, which makes it possible to retain the same solution form as in the case with only one stochastic factor, see McDonald and Siegel (1986). Thus, traditional studies addressing real options in a forestry context have applied a single-option approach. The current paper, however, must go beyond this as we consider the more complex case of an adjacency constrained harvest-planning problem. Here, harvesting one stand implies a postponement of the real option to harvest the other stand. Thus, the harvesting decision, and hence the real options which they represent, are—at least to some extent— exclusive. In this case, it is impossible to confine the problem to a single state variable. In fact, in a recent paper by Malchow-Møller and Thorsen (2003), the problem of choosing between entirely exclusive options is analysed, and they argue that this problem cannot be solved analytically. In the present paper, the work by Malchow-Møller and Thorsen (2003) is extended to the case where exclusiveness is not complete. This is exactly the case for the adjacency-constrained harvest problem, since after the restricted period of s years from harvesting the first stand, the forest owner is given the option to harvest the adjacent stand. Because of this complexity, we apply a rather stylised two-stands model to assess the costs of adjacency constraints and to derive the optimal harvesting rules under uncertainty and subject to the adjacency restrictions. As will be seen, we find that the extension is by far not trivial in terms of the results obtained and this represents a contribution in itself. Specifically, when assessing the costs of the adjacency constraint, we find that recognising and incorporating uncertainty increases the costs of the restriction. The larger the uncertainty and/or the restriction period, the larger the implied costs. While the effects of the restriction period on the costs are to be expected, the effects of varying the length of the restriction period on the optimal harvest rules are much more complicated as will be seen below. In the following section, we outline a simple continuous-time model. In Section 3, we describe the numerical technique used for obtaining a solution. Results are provided and discussed in Section 4, and conclusive comments made in Section 5.

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2. The model In this section, a general optimal-stopping problem for the continuous-time case with uncertainty is formulated. The model developed is highly simplified to allow for the use of numerical techniques to handle the implications of uncertainty. We consider a situation in which a private forest owner has two adjacent stands (of different species), and is given the option to harvest either one of the two stands first, at the cost of postponing the harvest of the second stand for at least some predetermined period, s. Since the harvesting of one stand implies at least a postponement of the harvesting of the other stand, the harvest options are to some extent mutually exclusive. Furthermore, since harvest returns are uncertain and the harvest decisions are irreversible, the problem possesses the key features of the type of stochastic control models found in the real-options literature. Thus, the decision problem has in principle an infinite time-horizon, but the problem terminates after the harvesting of the second stand. We assume that the present values of the harvest returns—including future returns from use of the land—evolve stochastically over time. More specifically, we assume that exercising the first harvest option costs CF and yields a (present value) return of F, whereas exercising the second option is associated with the cost CD and yields a return of D. It is assumed that CF and CD are positive constants, whereas F and D evolve according to the following Brownian motions: dF ¼ lF F dt þ rF F dz dD ¼ lD D dt þ rD D dx

and of ð1Þ

where lF and lD are the drifts and rF and rD are the variance parameters of the respective processes, whereas dz and dx are the increments of two Wiener processes, possibly correlated with E(dzdx)=qdt. The units of the cost variables can in principle be scaled to any particular area, but those used in the following can be considered as DKK/ha of forest. Note that there is no time-dependency in the values of the stands or other more complicated elements of a growth model involved. Although this is certainly a strong simplification, it is quite common in the literature, see, e.g., Reed (1993), Thomson (1992) and

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Zinkhan (1991). To ensure stationarity of the problem and allow for a practical solution, this simplification is needed. Furthermore, one can argue that in the case of mature stands, the age-dependent growth in timber value is often of a smaller magnitude than year-to-year variations in the present value of returns. In Scandinavia, for example, the growth in mature stands is often around 2 –4%, whereas timber price fluctuations easily reach F15 –20% a year. It is the effect of this kind of uncertainty on the costs of imposing adjacency restrictions and on the optimal timing of harvesting adjacent stands, which we explore. Now, assume that the forest owner maximises the expected net present value of his joint harvest options subject to the imposed adjacency restriction of at least s years between the harvests, and that he is able to observe at each point in time the states, F and D, of the two processes in Eq. (1). The optimal expected net present value, V( F, D), of this decision problem is then given by the following Bellman equation:

V ð F; 0Þ ¼ AF F bF

and

V ð0; DÞ ¼ AD DbD

ð3Þ

with associated stopping rules: hF CF hF  1 bD CD D*ð F ¼ 0Þ ¼ bD  1

F*ð D ¼ 0Þ ¼

and ð4Þ

where the problem specific constants, AF, AD, bF and bD, are given by:

V ð F; DÞ ¼  max F  CF þ eds E ½V ð0 D þ DDðsÞÞ; D  CD þ eds E ½V ð F þ DF ðsÞ; 0Þ; ð1 þ ddt Þ1 E ½V ð F þ dF; D þ dDÞg

where the options are completely exclusive. The opposite extreme case where s=0 implies no restrictions and therefore no exclusiveness. Before turning to the method and the results of the numerical study, we note that because zero is an absorbing state for either of the two state variables, F and D, the values V( F, 0) and V(0, D) are well known from the single-option case, see, e.g., Dixit and Pindyck (1994). They are given by:

AF ¼ ð2Þ

where d is the risk-adjusted discount rate, and DD(s) and DF(s) are the changes in the two Brownian motions over the finite time interval, s. The expression in Eq. (2) says that the optimal value is obtained by either immediately exercising one of the harvest options (the first two arguments in the curly brackets) or by continuing to keep both options alive for at least one more short time interval, dt, the value of which is given by the last argument, (1+ddt)1 E[V( F+dF, D+dD)] . The key difference compared to the problem analysed by Malchow-Møller and Thorsen (2003) lies in the value of stopping. In the present case, the value of stopping is not just given by the harvest return less costs, i.e. FCF or DCD. In addition to this, the forest owner receives, after a delay period of s years, the (single) option to harvest the adjacent stand. The present value of this option is given by edsE[V(0, D+DD(s))] or eds E[V( F+DF(s), 0)]. As s increases to infinity, these single options become worthless, and the problem, therefore reduces to the one treated by Malchow-Møller and Thorsen (2003),

F*  CF F * bF

and

AD ¼

D*  CD ð5Þ D * bD

ð5Þ

and: 1 l bF ¼  F þ 2 rF and 1 l bD ¼  D þ 2 rD

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   lF 1 2 2d  þ 2 >1 rF 2 rF ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s  lD 1 2 2d  þ 2 >1 rD 2 rD

ð6Þ

These functions are exact descriptions of the value function V( F, D) along the boundaries of the state space, and they are of course also the exact values of the single options received after the delay period, s. The key parameters here are those of the stochastic processes and the discount rate, since they determine the problem specific constants, AF, AD, bF and bD. For details of the theory behind this standard solution to the single-option problem, we refer readers to the book by Dixit and Pindyck (1994).

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3. The numerical approach For the problem to be solvable, the value function, V( F, D), must be finite. Since V( F, D) must necessarily be less than V(0, D)+V( F, 0)—the sum of the two individual option values—and since V(0, D)lD, and V( F, 0)lF, see, e.g., McDonald and Siegel (1986), it follows that V( F, D)max{lF, lD}. With a finite value function, the problem can be solved using a value-function iteration procedure, see, e.g., Judd (1998). To increase stability and iteration speed, the problem is log-normalised. The central gain from this is that the variance becomes independent of the state, thereby standardising the calculation of the joint probability distribution. The problem has been solved for a number of different parameter combinations. In this paper, however, we focus on an illustrative empirical case. In Thorsen (1999), the afforestation decision is investigated as a single real option, using empirical data on oak (Querqus robur L.) and spruce (Picea abies (L.) Karst.) prices. The empirical variance estimates from Thorsen (1999) are reasonable estimates for the present problem, as they consider the volatility of present values of stands on the stump and embed the variance of several assortment types using real Danish prices from 1911 to 1992. For oak, our F-variable, the standard deviation, rF, is 0.16 and for spruce, rD=0.13. The correlation of the present-value processes is estimated to qFD=0.49. The drifts in the present values, lF and lD, were not found to be significantly different from 0. Finally, it is assumed that the costs are: CF=CD=15 000 DKK/ha.

4. Numerical results We start by analysing the case which ignores the adjacency constraint. In this case, the harvesting of Table 1 Value function for four different time-delay constraints at three different states

V(10.2, 10.2) V(15.2, 15.2) V(20.1, 20.1)

s=0

s=10

s=20

s=100

1.72 5.14 11.08

1.14(66%) 4.16(81%) 9.70(88%)

1.09(63%) 3.85(75%) 8.68(78%)

0.94(55%) 3.09(60%) 6.61 (60%)

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Table 2 Deterministic value function for four different time-delay constraints at three different states

W(10.2, 10.2) W(15.2, 15.2) W(20.1, 20.1)

s=0

s=10

s=20

s=100

0.0 0.36 10.17

0.0 0.29 8.17

0.0 0.25 6.96

0.0 0.18 5.12

one stand does not affect the harvesting possibilities of the adjacent stand. This corresponds to s=0, and the optimal expected net present value, V( F, D), is just the sum of the values of the two separate options. This value can be expressed as:

(A F

V ð F; DÞ ¼ F

AF F

bF

þ AD DbD

if F < F* and D < D*

bF

þ D  CD

if F < F* and D z D*

bD

if F z F* and D < D*

F  CF þ AD D

F  CF þ D  CD if F z F* and DzD*

ð7Þ

where F* and D* are given by Eq. (4). Turning to the more general case, Table 1 compares the value function for four different time-delay constraints, s, at three different states, ( F, D). It is seen that the time-delay constraint affects the value function considerably. The stronger the restriction imposed, the larger is the cost in terms of a reduced value of the timber-harvesting options. Considering the state: F=D=10.2, the value of the unconstrained problem, s=0, is estimated at 1.72 decreasing to 0.94 when the time-delay constraint is set to 100 years. Note, however, that the loss caused by the time-delay constraint increases with s at a marginally decreasing rate. To illustrate the effects on the value function of incorporating uncertainty in the evolutions of F and D, Table 2 gives the value function, W( F, D), for the corresponding deterministic problem, where F and D are constants since the drifts are zero. As a consequence, the value function, W( F, D), equals zero in the state: F=D=10.2, since F=10.2
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With respect to the costs of the adjacency restrictions, we see from Table 1 that with uncertainty, positive costs can be identified even in the situations

where present value revenues are currently below harvest costs, see V(10.2, 10.2). The costs are also larger in absolute terms for the states where harvesting

Fig. 1. Panel a through f show stopping and continuation regions for six different time-delay constraints. Black areas are continuation regions, and grey areas are stopping regions. rF=0.16 and rD=0.13.

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revenues exceed harvesting costs. The difference, however, decreases as state values increase and immediate stopping, therefore becomes the optimal choice. To illustrate how the optimal timing of harvesting the first of the adjacent stands is determined in the stochastic model, Fig. 1 shows the shape of the continuation region, i.e. the values of F and D at which it is optimal not yet to harvest any of the two stands, and the two stopping regions, i.e. the values of F and D where it is optimal to harvest stand F or stand D. The regions are shown for six different time-delay constraints, s={0, 1, 10, 20, 100, and l}.1 For s=0, the two options are independent—there is no adjacency constraint. Hence, the stopping value of F is not affected by the value of D, and vice versa. This case is illustrated in Panel a of Fig. 1 (the top left corner), where the stopping boundaries are 24.75 in the F-direction and 22.56 in the D-direction. Thus, when F>24.75 and D>22.56, both options are exercised. Note how the larger variance of F implies a larger stopping value, see also Eqs. (4) – (6). For s=l, the two options become mutually exclusive (Panel f in Fig. 1). This corresponds to the case analysed by Malchow-Møller and Thorsen (2003). As D increases, the cost of exercising F increases, since you thereby forego the opportunity to switch to the D option, should it eventually become more profitable— a possibility, which becomes increasingly likely as D rises. Malchow-Møller and Thorsen (2003) show formally that at any point in the state space where DCD=FCF, it will always be optimal to continue holding both options alive for at least a short time interval—unless the two options are perfectly positively correlated. Hence, in Panel f of Fig. 1, the two stopping boundaries will never meet. Of course, a delay period of 100 years, or l for that matter, are not relevant in most real-world situations, but including these cases allows us to better illustrate the dynamics and complexity of the problem and furthermore provides a link to the work of Malchow-Møller and Thorsen (2003). For intermediate and more relevant values of s, the intuition is slightly more involved. On the one hand, 1 Following the delay period, s, after the harvesting of the first stand, the problem reduces to a standard single-option problem with optimal harvesting of the second stand given by the stopping rule in Eq. (4).

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the mechanism from the case where s=l is still present. As D increases, the cost of exercising F increases. This effect is of course most pronounced at large values of s, where you must forego the opportunity to exercise D for a longer period. On the other hand, the cost of continuing also increases with the value of the competing option, D, since continuing to hold the F option alive implies that the D option must also be kept alive for a longer period. This involves a cost in terms of lost interest rates on the net present value of the D option, which is received s years after exercising F. This cost increases with the value of D, but decreases with s, since a longer time delay implies a smaller value of the D option received after exercising F. Together, these two mechanisms can explain the pattern observed in Fig. 1. At low values of s, the effect of D on the cost of continuing dominates the effect on the cost of exercising and, therefore the stopping value of F decreases as D increases. At high values of s, the effect of D on the cost of exercising dominates, and hence the stopping value of F increases as D increases. Note, however, that the stopping value of F drops initially in Panels c and d in Fig. 1. This happens because the value of D only matters for the cost of exercising when D is sufficiently high to become a relevant alternative to F within the period s, whereas the cost of continuing increases steadily with D at all values of D. From Table 3, it is seen that the value function estimate increases as the correlation decreases from highly positive towards highly negative. Fig. 2 also shows that the effects on the stopping boundaries of this change are considerable. The stopping values increase as the correlation of the present value proTable 3 Value function at three different correlations and two different volatilities q=0

q=0.5

rF=0.16 and rD=0.13, V(10.2, 10.2) V(15.2, 15.2) V(20.1, 20.1)

s=20 1.09 3.85 8.68

q=0.49

1.12 4.03 9.03

1.12 4.14 9.34

rF=0.05 and rD=0.13, V(10.2, 10.2) V(15.2, 15.2) V(20.1, 20.1)

s=20 0.46 2.42 7.44

0.48 2.79 8.33

0.49 2.87 8.37

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Fig. 2. Sensitivity to changes in correlation between the two options and changes in volatility. Black areas are continuation regions, and grey areas are stopping regions. In Panels a – c: rF=0.16 and rD=0.13. In Panels d – f: rF=0.05 and rD=0.13.

cesses becomes increasingly negative. That negative correlation is good for the value of a batch or a portfolio of assets is a well-known result in economic

theory. It is also well documented empirically. The effect is attributed to the fact that the negative correlation decreases the variance of the portfolio yield.

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The decreased variance has a value, because the valuation is performed under the assumption of risk aversion. The current paper, however, values a portfolio of assets—real investment options—under risk neutrality. Thus, the value increase cannot be due to risk-reducing properties of the portfolio. Instead, it is entirely due to an improved probability of increasing the expected yield. The increased yield results from the fact that the expected maximum of two stochastic assets is never smaller than their means, see also Malchow-Møller and Thorsen (2003). In the lower part of Table 3, the effects of decreasing the variance of one asset (the F asset) are shown. Less volatility decreases the value function, as in a standard single-option model. The continuation region also contracts as visualised in Fig. 2—not only for the F option (as in the single-option case), but also for the D option. The incentive to keep the D option alive is smaller, since the lower volatility of F makes it less likely that exercising the F option will become the better choice in the near future. This is particularly relevant when F is high and/or the correlation is negative.

5. Concluding remarks The logging of larger tracts of forest land is often found to be detrimental to many environmental values as well as social values. For these reasons, constraints are often imposed upon harvesting options on adjacent forest stands. As is often the case with management restrictions, they commonly inflict costs upon the private forest owners in terms of revenues foregone and a decreased market value of harvesting. These costs and the resulting optimal harvesting plans have so far only been evaluated using deterministic models. In this paper, however, we explore the effects of uncertainty on the costs of adjacency restrictions in a rather stylised two-stands model. Furthermore, we examine aspects of the optimal harvest rules under adjacency restrictions and uncertainty. This is accomplished by extending the real-options theory to reflect this rather complicated stochastic control problem of more or less mutually exclusive real options. The model builds on the two-options approach developed by Malchow-Møller and Thorsen (2003).

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It is shown that introducing the adjacency constraint affects the value function considerably in a negative direction. As the delay constraint increases, the loss increases at a marginally decreasing rate. This of course has the policy implication that the expected benefits to society from imposing adjacency restrictions should be evaluated relative to a larger cost than typically estimated using deterministic models. Note that this does not necessarily imply that sustainability improving measures should be undertaken less frequently. The benefits and options to society associated with the restrictions are also often assessed using deterministic methods, which may in many cases also imply an underestimation, see Dixit and Pindyck (1994). While this remains an interesting topic for future research, it is beyond the scope of the present paper. Apart from the costs related to the imposed adjacency restrictions, the consequences for the optimal harvesting rules (which are of course also the costminimising harvesting rules) are of interest. These are evaluated for a number of different time delays revealing some interesting findings. First, in the case of a strong time-delay constraint, the results correspond to those of Malchow-Møller and Thorsen (2003), where the treatment of exclusiveness is similar to an infinite time delay. In this case, we see that the stopping rules, F*(D) and D*( F), increase monotonically and in a convex fashion in their arguments. Second, without a time-delay constraint, the options can be treated separately, and the stopping rules F*(D) and D*( F) are thus independent of D and F, respectively. Third, at intermediate time-delay constraints, the stopping rules reflect two opposing forces. On the one hand, the cost of exercising increases with the value of the opposing option, but on the other hand, the cost of continuation also increases. The latter effect dominates at small time-delay constraints, and it causes the stopping rules to initially decrease in their arguments.

Acknowledgements The authors are grateful for the valuable comments from Dr Shashi Kant and two anonymous referees. These comments helped improve the paper significantly. The authors also wish to thank Jens Iversen for excellent research assistance.

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