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Estimating the costs of overlapping tenure constraints: a case study in Northern Alberta, Canada
Forest Policy and Economics 8 (2006) 610 – 624 www.elsevier.com/locate/forpol
Estimating the costs of overlapping tenure constraints: a case study in Northern Alberta, Canada David M. Nananga,*, Grant K. Hauerb a
Natural Resources Canada, Canadian Forest Service, 580 Booth Street, Ottawa, Ontario, Canada K1A 0E4 b Department of Rural Economy, University of Alberta, Edmonton, Alberta, Canada T6J 0N6 Received 18 March 2004; received in revised form 1 November 2004; accepted 10 November 2004
Abstract The main objectives of this study were to: (i) incorporate the types of constraints implied by overlapping tenures into a timber supply model; and (ii) estimate the costs associated with various constraints implied by overlapping tenures in Alberta, Canada. This was achieved through a linear programming formulation of the timber supply problem over a 100-year planning horizon, which resulted in more than 5 million decision variables and approximately 118,000 constraints. Given the size of the model, a dual-decomposition procedure to solving large-scale linear programming problems was applied. The results showed that, in general, constraints imposed by overlapping tenures led to inefficiencies in wood allocation and substantial increases in the marginal costs of production. Secondly, the effect of the overlapping tenure constraints was unevenly distributed among mills. The costs to individual tenure holders are highly dependent on how far the mills are from their allowed harvest locations and how far the constraints shift mill harvest areas away from their optimal wood procurement zones. Removal of the constraints leads to a 7% increase in the net present value of the forest. For mills that are located within short distances of their allowed harvest locations, removal of constraints do not significantly lower marginal costs. The results show that overlapping tenure constraints are inefficient and therefore should be removed and better ways of allocating land for harvest should be sought. Although such policies would be efficient, tenure holders who derive an economic advantage from existing arrangements will oppose them. In these cases, some means of compensating mills that lose as a result of more efficient wood allocation may have to be arranged. D 2004 Elsevier B.V. All rights reserved. Keywords: Decomposition techniques; Forest management scheduling; Linear programming; Overlapping tenures; Shadow prices
1. Introduction
* Corresponding author. Tel.: +1 613 947 9023; fax: +1 613 947 9020. E-mail address: [email protected] (D.M. Nanang). 1389-9341/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.forpol.2004.11.003
Changing technology in wood processing in Alberta, Canada has made it possible to utilize species of wood, which previously had no commercial value. The emergence of these species as valuable resources
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
occurred while traditionally valued species often growing as part of the same forest stand also increased in value (Luckert, 1991). This situation led to new management problems associated with the joint production of multiple outputs from forestland. The Alberta Forest Service has been allocating harvesting rights to more than one firm on the same piece of land as one way of ensuring that the newly valued resources are utilized (Luckert, 1991). As a result, overlapping tenures or divided land bases have emerged in Alberta as a significant issue as the number of forest management agreement (FMA) areas increased. The usual configuration of overlapping tenures in Alberta is that one firm has an area-based tenure with rights to harvest over the whole area of the forest as well as rights to harvest all or most of the tree species on the areas. These rights carry obligations for the firm to regenerate harvested areas; plan the sequencing of harvests; and ensure that harvesting operations are sustainable. The full set of rights and obligations is set out in an FMA between the timber firm and the provincial government. The term boverlapping tenureQ comes about when, within the FMA areas, there are embedded volume-based tenures, which are held by other firms. Although the volume-based tenures or quotas are not barea-basedQ like FMAs, quota holder harvests are usually spatially restricted to specific areas within FMA boundaries. In addition, volumebased tenures are usually species-specific. Only certain species may be harvested by the quota holders. To complicate matters further, forestland is usually classified by predominant species and quota holder harvesting rights are usually restricted to the part of the land base where the species specified by the quota predominates. For example, if the volume quota is for conifer, the quota holder may be restricted to harvesting off the part of the land base classified as conifer. In some cases, quota holders may also have rights to incidental volumes harvested off other land bases. Overlapping tenure constraints also interact with other regulations. First, regeneration standards are specified to return forest stands to approximately the same species composition that existed before harvest. Hence, conifer stands are regenerated to return to conifer, deciduous to deciduous, and mixed to mixed stands. This may in some cases prevent stands from
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being regenerated most cost-effectively from a social perspective. Second, there are often implicit sustainability constraints such as annual allowable cut (AAC) constraints (or even-flow constraints) applied to the land bases, embedded within the larger FMA land base, from which quota holders draw their wood supply. Finally, it is usually unclear as to who has rights to increased allowable cuts that may be obtained by increasing silvicultural input into the forest (the so-called allowable cut effect). A few researchers have previously examined overlapping tenures in Alberta. Cumming and Armstrong (1999) used a simulation approach to conduct a quantitative examination of the costs of overlapping tenures and divided land bases in Alberta. They concluded that the costs of overlapping tenures and divided land bases are substantial enough to justify a thorough examination of forest policy regarding overlapping tenures in Alberta. One other study by Alavalapati and Luckert (1997) modeled the short-run timber supply of quota holders in Alberta in the face of institutional constraints (allowable cut and mill capacity) and fixed stumpage prices using dynamic optimization techniques. The shadow prices of mill processing capacity and allowable cut restrictions were estimated for large, medium, and small tenure holders to reflect the different cost structures of different-sized firms. The results indicated that all categories of quota holders studied incurred substantial costs due to these two institutional constraints, and that simultaneous elimination of both constraints leads to more cost reduction than the combined savings from eliminating each constraint individually. The focus of that study was on quota holders, and therefore did not address the special problems related to overlapping tenures involving FMA holders that are investigated in this study. Despite the apparent inefficiencies in the application of overlapping tenures, the situation persists because quotas are largely historical and politically entrenched rights to species that, in many cases, were granted before the current FMAs were signed. The rights exist as a volume allotment in areas where the desired species exist. Therefore, a key problem is that there is a lot of inertia to reallocate, as many of these small operators have roots in the locations where they operate.
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The objective of this paper is to estimate the costs associated with various constraints implied by overlapping tenures for a case study area in Alberta. Our approach is to formulate a mathematical program-
ming problem for the situation and estimate the costs by incorporating constraints and then relaxing them to represent policy change. Incorporating the constraints implied by overlapping tenures complicates
Table 1 Summary of mill target demands and overlapping tenure constraints Model run
Demand (Eq. (7))
Restrictions on areas that mills are allowed to harvest (Eq. (6))
Demand sites
Demand location
Max mill price ($/m3)
Max mill demanda
Wood type
Allowed locations
Allowed stand types
B A C B A C B
85 85 85 65 65 60 85
220 20 20 215 200 100 220
SW SW SW HW HW SW SW
All
All
Scenario 1
Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1
SW, HW
Scenario 1a
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1
A C B A C B
85 85 65 65 60 85
20 20 215 200 100 220
SW SW HW HW SW SW
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6l Sawmill 1
A C B A C B A C B A C B
85 85 65 65 60 85 85 85 65 65 60 85
20 20 215 200 100 220 20 20 215 200 100 230
SW SW HW HW SW SW SW SW HW HW SW SW
South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1 and 2 North FMA subregion 3 All All North FMA subregion 3 South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1, 2 North FMA subregions 3 All All North subregion 3 All
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6
A C B A C B A C B A C
85 85 65 65 60 85 85 85 65 65 60
22 22 240 220 110 230 22 22 240 220 110
SW SW HW HW SW SW SW SW HW HW SW
Baserun
Scenario 1b
Scenario 2
Scenario 3
SW=conifer; HW=deciduous. Subregions refer to FMUs within FMAs. Prices and costs are in 2003 Canadian dollars. a Maximum volume demand per year for the mill in cubic meters.
South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1, 2 North FMA subregions 3 All All North subregion 3 All
SW SW HW, SW HW, SW SW All
SW, HW SW SW HW, SW HW, SW SW SW, HW SW SW HW, SW HW, SW SW All
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model formulations in several ways. First, to properly represent the overlapping tenure situation, more than one demand location must be represented so that the different tenure holders may be modeled. To fully capture the costs of constraints, which are spatial both because they are applied over forest management units (FMUs) within an FMA, and because of the different demand locations, multiple supply locations must be modeled. The policy to allow overlapping tenures, which seek to guarantee a supply of timber to quota holders over a long period of time, makes it necessary to test the sustainability and cost of the existing wood supply configuration over a long period of time. Furthermore, since overlapping tenures impose restrictions on where and on what kind of land class wood can be taken, models must keep track of which land classes and which locations desired volumes are being taken from. This is because sawmills and pulpmills use only conifer species, while OSB mills use deciduous species only (Table 1). These temporal and spatial requirements considerably increase the number of decision variables as well as constraints to be modeled. The timber supply model is formulated for two FMA areas in northern Alberta, which have significant overlapping tenure situation. While the description of the land base and the overall overlapping tenure situation are realistic, the model implementation is modified enough to be less realistic in terms of the magnitude of the direction of changes, the wood demands represented, and the number of mills represented. The overlapping tenure constraints that are explicitly examined in this paper are harvest location and predominant species land class restrictions for quota holders. 1.1. Model formulation Two FMA areas in Alberta with a total productive forest area of about 550,000 ha were used in this study. There is one FMA area in the north and one in the south (north and south FMA, respectively) and the spatial units of analyses were 2500-ha grid cells (see Fig. 1). Mathematically, the timber supply problem can be described by the set of equations given in Eqs. (2– 7). The formulation is an extension of the Model II formulation given in Johnson and Scheurman (1977).
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Fig. 1. The study area showing the north and south FMA areas and the three demand locations.
The objective function (Eq. (1)) maximizes the net benefit of timber products from all mills (firms) with rights to timber on the two FMAs subject to the standard Model II age class constraints, demand constraints, and overlapping tenure constraints. The benefits are the net returns of the value of wood products plus the value of ending forest inventory minus costs of regeneration, harvesting, and transportation. The first term in the objective function (Eq. (1)) is the discounted value of wood products. While our model is general enough to handle downwardsloping demand curves, simplified demand curves are used here, which are perfectly elastic (single price) up to a maximum mill capacity. The second term in Eq. (1) is the value of the ending inventory, whilst the last term is the costs of regeneration, harvesting, and transportation: M X T X
max
t1
"Z
b
m¼1 t¼1
þ
Ji X I T X X
#
ymt
Dmt ð X ÞdX
0
Eins wins
i¼1 j¼1 t¼1
Ji I tz X T X X X i¼1 j¼1 s¼T 0 t¼1
cijst xijst
ð1Þ
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Ji X T X
xijst þ
j¼1 t¼1
Ji X
8i; s ¼ T 0 ; N ; 0
wijs ¼ Ais
j¼1
Ji T X X
xijth þ
j¼1 h¼tþz
Ji X
wijt
j¼1
Ji tz X X
ð2Þ xijst ¼ 0
j¼1 s¼T 0
8i; t ¼ 1; N ; T
ð3Þ
land base regeneration restrictions: ! ! T J T X X X X c xijst þ wijs zd xijst þ wijs jaJic
t¼sþz c
cd
8iaI ; I ; I ; T X
jaJid
t¼sþz
8iaI d
8s ¼ 0; N ; T z !
X
t¼sþz
j¼1 dc
xijst þ wijs zd
d
ð4Þ
J X
T X
j¼1
t¼sþz
! xijst þ wijs
8s ¼ 0; N ; T z
ð5Þ
allowable harvest area and land base restrictions (LBRs): XX vijstm xijst V0 8s; t; and m ð6Þ iaI m jaJim
market clearing restrictions: Ji I tz X X X
vijstm xijst Vymt
8t; m ¼ 1; N ; M
ð7Þ
i¼1 j¼1 s¼T 0
xijst z0
8ijst
wijs z0
8ijs
ymt z0
8mt
Area of stand type i managed with regeneration prescription j, in period s and left as ending inventory c ijst The discounted cost per unit area of managing stand type i with regeneration and market shipping plan j, starting in period s and final harvest in period t x ijst Area managed on stand type i with regeneration prescription and market shipping plan j, starting in period s and final harvest in period t v ijstm The volume per unit of wood products from mill m, in period t, when stand type i is regenerated in period s and managed with prescription and market shipping plan j y mt Quantity of wood demanded by mill m in period t dc Percentage of analysis area that must be regenerated to conifer species dd Percentage of analysis area that must be regenerated to deciduous species Ji The set of regeneration prescriptions and transport destinations for analysis area i; it should be noted that wood from any stand i can be sent to more than one destination Jci Subset of J i that includes regeneration prescriptions that meet the conifer standards Jdi Subset of J i that includes regeneration prescriptions that meet the deciduous standards I c, I d, I dc, I cd Set of conifer, deciduous, mixed deciduous/conifer, and mixed conifer/deciduous land bases, respectively I The number of stand types Im The set of locations or forest types that are not available to mill m Jmi The set of transport/prescriptions that are defined for mill m z Minimum time between regeneration and harvest T The number of planning periods in the planning horizon b Discount factor (using 5% discount rate). w ijs
subject to Model II forest dynamics constraints:
where: D mt (X) The inverse demand function for wood from mill m in period t A is The number of area unit of stand type i in the first period that were regenerated in period s E ijs The discounted value per unit area of managing stand type i with regeneration prescription j, starting in period s and leaving the stand type as ending inventory
Eqs. (2) and (3) are standard Model II age class dynamics constraints (see Johnson and Scheurman, 1977). These equations ensure that total area harvested does not exceed the initial size of the land base and that all areas harvested beyond the first cut were harvested and regenerated previously. Current provin-
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cial regulations require that conifer land bases be regenerated to predominantly conifer and deciduous land bases to predominantly deciduous species. Constraints (4) and (5) represent these additional regulations. Eq. (6) represents restrictions on where quota holders are allowed to harvest. These constraints can incorporate both spatial restrictions that confine harvest to specific subareas within the FMA area (harvest location restrictions (HLRs)) and restrictions on what type of land base (conifer, mixed conifer, mixed deciduous, and deciduous) may be harvested (LBRs). Eq. (7) implies that volume of wood products produced from all stands managed in the two FMAs should be less than or equal to the mill demands. There are no AAC constraints in this model because AAC is assumed to be highly correlated with mill capacity (maximum mill demand). Increases in mill demand are assumed to drive increases in AAC or, conversely, any increase in AAC is assumed to be absorbed by increases in mill demand or capacity. Hence, in this model, AAC regulations are embedded in the demand structure and are essentially represented by the summation of the mill demand constraints. While Eqs. (2–7) set out the economic problem, the enormous size of the formulation will be difficult to solve, especially if a nonlinear demand system is specified. The model specification for the case study area has 29,885 forest type/location/age class combinations; a 100-year planning horizon with 10 planning periods; a minimum rotation of 40 years; three regeneration prescriptions per stand; a total of 6741 stand types identified by species types, site class, and location; and approximately 10 shipping alternatives for each stand.1 With these assumptions, the resulting model has approximately 5 million decision variables and about 118,000 constraints. Given the difficulty of solving a model of this size, a variant of the dual-decomposition
1
This is based on three species with two size classes for each species, four possible destinations for softwood species, two possible destinations for hardwood species, and the assumption that only half of these alternatives would be available for each stand on average.
615
algorithm proposed by Hoganson and Rose (1984) and later modified by Hauer and Hoganson (1996) was used to solve the problem. Some details of this approach are included in the appendices. 1.2. Model scenarios and overlapping tenure constraints Six hypothetical scenarios were developed to assess the costs of overlapping tenure constraints under different conditions. The scenarios are summarized in Table 1. The Baserun is meant to represent a case where there are no overlapping tenure restrictions. In the Baserun, the model allocates the wood across mills so as to maximize overall net benefits. Scenario 1 is designed to reveal the joint effect of HLRs and LBRs relative to the Baserun, which has no such restrictions. In Scenario 1a, the LBRs are removed, leaving only the HLRs. Scenario 1b drops the HLR and examines the effect of the LBRs only. In Scenario 2, maximum mill demands are assumed to absorb an increase in AAC. The increased AAC is distributed proportionately to all tenure holders, based on their shares of the AAC for each FMA. Scenario 3 uses the same AAC allocations as Scenario 2, but Scenario 3 drops the overlapping tenure constraints. Scenarios 2 and 3 are used to simulate the effect of changes in the AAC and the joint effect of overlapping tenure constraints and changes in AAC on changes in marginal costs of timber production. The second column in Table 1 identifies the demand sites of which there are six. We shall refer to demand locations as A, B, and C. Each demand location has two demand sites, which include sawmills, OSB mills, and pulpmills. Sawmills 1, 2, and 3 are located in demand locations B, A, and C, respectively. OSB Mills 4 and 5 are located at demand locations B and A, respectively. The pulpmill is located at location C. The fourth and fifth columns of Table 1 are sufficient to describe the simplified demand curves for three end products used in this analysis, with demand being perfectly elastic at a maximum price (column 3) and perfectly inelastic along a vertical segment (column 4). The third column shows an estimate of the maximum price in terms of dollars per cubic meter of final product that could be paid at the mill gate. These
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maximum price levels were set based on current estimates of the prices of lumber and OSB. For the pulpmill, the final product is defined as an intermediate product: wood chips. The fourth column shows the maximum volumes that can be consumed by each mill on an annual basis (the right-hand side of Eq. (7)). The maximum volumes for hardwoods and softwoods add up approximately to the total volumes of these species harvested on the north and south FMA areas during the last 5 years. They also add up approximately to the allowable cuts. Hence, wood demands of mills from this area are roughly equivalent to allowable cuts levels. The breakdowns of volumes by mill are based on the percentage shares that each mill is allocated under the existing FMA and quota agreements. The last two columns in Table 1 summarize the overlapping tenure constraints (Eq. (6)). The second to the last column shows the allowed locations from which each mill can harvest (HLR), while the last column shows the cover types from which each mill is allowed to harvest (LBR). As pointed out earlier, these restrictions in harvest locations are a result of the historical development of overlapping tenures and do not necessarily reflect a desire to optimize overall net benefits from timber harvest. Sawmill 1 in location A and the two OSB mills represent the FMA tenure holder’s demand locations. As part of the tenure agreements, the OSB mills may harvest from anywhere in the two FMA areas. However, in Scenarios 1, 1a, and 2, Sawmill 1 is limited to harvest in the south FMA and only parts of the northern FMA, while the other sawmills and the pulpmill may harvest only from specified portions of the northern FMA. In the Baserun and Scenario 3, these constraints are eliminated entirely so that wood may flow to any mill from any location.
2. Results and discussion Net present values for each scenario are presented in Table 2.2 Comparisons of the Baserun and Scenarios 1, 1a, and 1b show that the Baserun has
2 All prices and costs are in 2003 Canadian dollars. A real discount rate of 5% was used.
Table 2 Net present values for the six scenarios Model run
Objective function value (109 $)a
Baserun Scenario Scenario Scenario Scenario Scenario
2.311 2.140 2.142 2.154 2.302 2.477
1 1a 1b 2 3
a Values are in 2003 Canadian dollars, discounted with a 5% real discount rate.
the highest NPV as expected. The total loss in NPV due to the constraints in Scenario 1 as compared to the Baserun is $171 million, which represents a loss of about 7% of the objective value. However, having both constraints in place only marginally increases the cost of the constraints. From Table 2, imposing HLRs alone decreases NPV by $169 million. Imposing LBRs on top of HLRs decreases NPV by only $2 million. Similarly, imposing LBRs alone decreases NPV by $157 million, whilst imposing harvest area restrictions on top of the LBRs only decreases NPV by $14 million. Shadow price estimates on mill demand constraints, which may be interpreted as marginal costs of production, for the Baserun and Scenario 1 reveal a number of important patterns (Figs. 2–4). First, we describe some of the general patterns. Marginal costs for the Baserun tend to increase over the planning horizon for the sawmills, decrease for the OSB mills, and stay flat or increase for the pulpmills. The increased marginal costs observed are due to long-term scarcities that emerge as more of the wood is harvested off the FMAs. Since the Baserun does not contain any overlapping tenure restrictions, the only reason for increasing marginal costs is scarcity of wood over time, forcing firms to harvest from poorer sites. Shadow prices for OSB mills for the Baserun remain the same or slightly decrease over the planning horizon (Fig. 2). This reflects a relative abundance of aspen wood on the two FMAs. Another pattern that can be seen in Fig. 3 is that marginal costs for all sawmills tend to be closer together when overlapping tenure constraints are relaxed than when they are applied. This occurs
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Fig. 4. Comparison of shadow prices for pine and spruce chips for the Baserun and Scenario 1. Fig. 2. Shadow prices for OSB mills for the Baserun and Scenario 1.
Fig. 3. Comparison of the shadow prices for lumber mills for the Baserun and Scenario 1.
because the only major difference in costs in our model that can exist once overlapping tenure constraints are removed is in transport costs to the mills. At the margin of each mill’s woodshed, the value of sending wood to the competing mills will be roughly equivalent, which agrees with the equimarginal principle. Overlapping tenure woodsheds are not determined by economic criteria and hence we see a divergence in marginal costs. We now turn to the effects of overlapping tenure constraints on marginal costs of production. Marginal costs for Sawmill 3 are significantly higher when overlapping tenure constraints are imposed than in the Baserun. For Sawmill 2, marginal costs are higher when overlapping tenure constraints are imposed except in periods 7 and 10, when they are roughly equivalent. However, for Sawmill 1, the marginal costs, while higher in the first four periods, actually decrease below the marginal costs in the last six periods when there are no overlapping tenure constraints. The overlapping tenure constraints restrict some mills’ harvesting to certain locations and land bases. Hence, one expects costs to drop
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once the constraints are removed. The marginal costs for Sawmill 3 are higher with the constraints because the locations that this mill is allowed to harvest from are outside its woodshed if no constraints were imposed. These results show that although dropping the constraints results in an overall increase in net returns and reductions in costs, the gains are not evenly spread across mills and over time. In this
case, for Sawmill 1, marginal costs are actually lower in the long run when there are overlapping tenure constraints. Insights into why the marginal costs change in the direction they do can be gained by examining the wood procurement zones for each mill. The wood procurement zones for Sawmill 1 are shown in Fig. 5a and b. In the Baserun, without overlapping tenure
Fig. 5. Wood procurement zones for Sawmills 1 and 2 for the whole planning horizon.
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
constraints, Sawmill 1 takes wood from locations that are closer to the mill compared to when there are restrictions. When there are restrictions, harvests for Sawmill 1 spread over a larger area and, in particular, they spread into the northern parts of the northern FMA area. Marginal costs increase for Sawmill 1 in the short run because Sawmills 2 and 3 are forced to harvest in areas where it is more profitable for Sawmill 1 to harvest, thus pushing Sawmill 1’s harvest into less profitable areas. However, Sawmill 1’s marginal costs eventually decrease when restrictions are imposed because, in the long run, there is more wood available to Sawmill 1 because Sawmills 2 and 3 are restricted to small areas within the FMA area. The optimal harvest locations (i.e., without HLR and LBR) for Sawmill 3 are shown in Fig. 6a, which is dramatically different from the woodshed with overlapping tenure constraints shown in Fig. 6b. Part of Sawmill 3’s increase in marginal costs will be due to increased transport costs and part will be due to lack of long-term flexibility and scarcity that result from being constrained to harvest only in the areas shown in Fig. 6b. Similar results for Sawmill 2 can be observed by examining Fig. 5c and d. Fig. 4 shows the marginal costs for softwood chips or pulpwood, and Fig. 6c and d shows the wood procurement zones with and without overlapping tenure restrictions. Removal of the overlapping tenure constraints results in a decrease in marginal costs in all planning periods. The woodshed without the constraints shows that wood destined for the pulpmill is harvested from almost every part of the two FMAs. The reason for the differences in the wood procurement zones appears to be related to wood sorting. There are efficiencies built in the model so that stands can be scheduled to deliver product to multiple mill destinations. When there is no constraint on wood harvesting, wood is taken from many places in the management unit because pulpwood components of conifer stands are shipped to the pulpmill and sawlogs are shipped to sawmills (not necessarily to the same location or tenure holder). When the constraints are put in place, the pulpwood that could have been delivered from the north of Demand Centre A (see Fig. 6c) (which would have been efficient) can no longer be delivered under the constraints. All the pulpwoods
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must be delivered from within the narrowly defined supply area of Fig. 6d. 2.1. Marginal costs of overlapping tenure constraints Table 3 shows the shadow prices or marginal costs of the overlapping tenure constraint set given by Eq. (6). All three sets of shadow prices are described by the same constraint except that the set over which the summation takes place is different in each case. The first set is a combination of restrictions on harvest area and land bases where mills can take their wood. The second is the harvest area restrictions alone, whilst the third is the LBRs. These shadow prices represent the marginal reduction in the objective function value as a result of not allowing a small amount of wood to be harvested from the areas (or land base) from which the mill is restricted from harvesting. Hence, since the shadow price for Sawmill 2 is $3.62/m3 in the first period, the overall objective function would increase by $3.62 for every cubic meter of wood that it could harvest from outside the area from which it is currently allowed to harvest. An examination of Table 3 reveals that for the combined harvest area and LBRs, the shadow prices on Sawmill 3 and the pulpmill (both in demand location C) are higher than the other mills. This is because the FMU allocated for harvesting to these two mills is far outside the two mills’ woodshed if there were no restrictions. In contrast, Sawmills 1 and 2 are located close to the FMUs that they have harvest rights to (compare Fig. 5c and d with Fig. 6a and b), and so the marginal costs of the overlapping tenure constraints are lower. The shadow prices on the harvest area restrictions are much larger than those of LBRs. This occurs because the LBRs prohibit a mill from harvesting deciduous land base if it is a conifer mill and vice versa—so mills are being prevented from harvesting wood from areas that they are less likely to want to harvest in the first place. Hence, the harvest area restrictions are more binding than the LBRs. 2.2. Effect of increased AAC on NPV and marginal costs Next, we consider the effect on marginal costs of the three final products following an increase in
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Fig. 6. Wood procurement zones for Sawmill 3 and Pulpmill 6 for the whole planning horizon.
AAC with and without overlapping tenure constraints. Table 4 shows the average 10-year differences in marginal costs with and without overlapping tenure constraints for all six mills. As pointed out earlier, raising the AAC requires investment from the firms over and above what is required by regulation, however, there is no clear mechanism
for distributing the increased AAC among the tenure holders. In this study, we assumed that the increased AAC is distributed proportionately to all tenure holders, based on their shares of the AAC for each FMA. A comparison of the difference in marginal timber costs in Scenarios 2 and 3 to the difference in
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Table 3 Final shadow price estimates of the harvest area and land base restrictions for Scenario 1 by period Constraint type
Planning period 3
4
5
6
7
8
9
10
Harvest location and land base restrictions Sawmill 1 3.62 4.73 Sawmill 2 6.93 6.45 Sawmill 3 6.58 6.92 Pulpmill 6 15.35 13.85
1
2
9.62 8.56 10.27 15.37
9.36 8.88 11.16 18.97
13.15 13.03 17.79 20.78
15.67 16.12 22.89 26.94
21.96 23.23 33.92 39.59
18.69 20.88 35.83 37.96
22.10 23.01 43.73 46.32
22.39 23.32 47.88 47.88
Harvest location restrictions Sawmill 1 3.06 Sawmill 2 6.93 Sawmill 3 6.58 Pulpmill 6 15.35
4.28 5.67 7.20 12.89
9.41 9.44 10.67 15.37
8.49 7.99 10.30 18.14
13.70 13.30 17.16 18.87
16.73 16.58 21.57 24.33
22.30 22.09 29.98 33.38
23.68 23.51 27.10 32.00
26.62 27.10 27.29 38.52
27.13 26.09 32.27 37.15
Land base restrictions Sawmill 1 0.00 Sawmill 2 6.25 Sawmill 3 2.97 Pulpmill 6 7.49
0.00 1.55 1.53 7.34
0.00 1.03 0.00 9.49
0.00 6.75 0.06 11.87
0.00 6.72 1.17 14.15
0.00 1.37 0.16 9.02
0.00 2.81 0.21 11.56
0.00 5.53 0.32 13.23
0.00 5.57 3.42 22.34
0.00 0.59 0.15 10.61
No OSB results are shown because no harvest location or land base restrictions were placed on the OSB mills.
marginal costs for Scenario 1 and the Baserun shows that overlapping tenure constraints have a greater effect when allowable cut levels are higher. With the exception of Sawmill 1 and OSB mill 1, the marginal costs of the demand constraints all increased by a greater degree when overlapping tenure constraints were present (compare rows 1 and 2 in Table 4). Another way of looking at the costs of the overlapping tenure constraints is to examine the change in marginal costs of production when allowable cut is increased with and without the constraints. Results in rows 3 and 4 of Table 4 show that when allowable cut is increased, the marginal cost increases are much lower when there are no overlapping tenure constraints present. Increasing cut levels is less expensive when the constraints are not present.
It is important to point out that changes in the assumptions of the model will affect the conclusions and management implications for tenure holders. For example, we assume a 100-year planning horizon during which the tenure holder abides by all the requirements of the tenure agreement, which is not quite realistic. It is possible that, over time, the tenures could be renegotiated with different allocation of rights that would reduce the costs of the overlapping tenure constraints. For the three products that were modeled, an increase in the number of OSB mills, for example, could lead to more deciduous species being harvested, and hence an increase in the costs of the constraints. Increases in the prices of final products will increase profitability for the firms, although it is unlikely to affect the costs of the constraints imposed by overlapping tenures.
Table 4 Average 10-year increase in marginal costs ($/m3) between selected scenarios Differences Scenario Scenario Scenario Scenario
1 2 2 3
minus minus minus minus
Baserun Scenario 3 Scenario 1 Baserun
Sawmill 1
Sawmill 2
Sawmill 3
OSB mill 1
OSB mill 2
Pine chips
Spruce chips
0.736 3.96 1.32 0.57
0.43 1.71 1.62 0.57
15.87 33.61 18.23 0.05
8.05 7.90 0.11 0.03
7.56 7.61 0.35 0.24
21.16 38.34 20.94 3.69
30.37 45.57 15.90 0.55
622
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
3. Conclusions This study applied an optimization approach to estimate the cost of overlapping tenure constraint on FMA areas in northern Alberta. Overlapping tenures continue as a policy instrument in Alberta because quotas are largely historical and politically entrenched rights to species that, in many cases, were granted before FMAs were signed. The results from the various scenarios investigated showed that overlapping tenure constraints are costly. The costs to individual tenure holders are highly dependent on how far the mills are from their allowed harvest locations and how far the constraints shift mill harvest areas away from their optimal wood procurement zones. Although, in general, removal of the constraints leads to decreased costs, the benefits of removing constraints are unevenly distributed among tenure holders. Removal of the constraints leads to a 7% increase in the net present value of the forest. For mills that are located within short distances of their allowed harvest locations, removal of constraints does not significantly lower marginal costs. Increases in AAC are more costly when overlapping tenure constraints are present than when they are removed. The results here suggest that overlapping tenure constraints should be removed and better ways of allocating land should be sought. One possible policy solution would be for the Alberta Forest Service to grant tenure holders comprehensive rights to all species within a licensed area (Luckert, 1991). Tenure holders could negotiate with each other to trade and/or sublease their rights to certain species that cannot be utilized in their mills or are in locations that are too far away from their mill locations. This would likely push the wood allocation closer to that of the Baserun as wood would tend to find its highest valued use. While the results here show overall net benefit improvement from removing overlapping tenures, they also suggest that, in some cases, removal of overlapping tenure constraints may decrease flexibility for some mills, resulting in increases in costs for those mills. Although policies to remove overlapping tenure constraints would be efficient, tenure holders who derive an economic advantage from existing arrangements will oppose them. In these cases, some means of compensating mills that lose as
a result of more efficient wood allocation may have to be arranged.
Appendix A. The dual-decomposition approach: theory We start with the Lagrangian function for Eqs. (2)– (4) and Eqs. (6)–(7): M X T X
min Lða; ar ; k; lÞ ¼ max r x;y;w
a;a ;k;m
"Z
#
ymt
m¼1 t¼1
Dmt ð X ÞdX þ
0
Ji X I T X X
cijst xijst þ
"
Ais
Ji X T X j¼1 t¼1
Ji X tz X
t¼1 s¼T 0 m¼1
þ
#
wijs þ
Ji T X X
"
lmt
t¼1 m¼1
"
ktm
ais
I T X X
arit
i¼1 t¼1
xijth
j¼1 h¼tþz
T 0 M X X X
T X M X
Ji X j¼1
xijst
j¼1 s¼1
þ
xijst
I 0 X X i¼1 s¼T
i¼1 j¼1 s¼T 0 t¼1
Eins wins
i¼1 j¼1 t¼1
Ji I tz X T X X X
"
bt1
Ji X
#
wijt
j¼1
XX
#
vijstm xijst
iaI m jaJim Ji I tz X X X
#
vijstm xijst ymt
i¼1 j¼1 s¼T 0
ðA1Þ Our solution approach proceeds by relaxing the requirement for strict feasibility of Eqs. (6) and (7) and assuming that we have knowledge of the shadow prices on these constraints, k mt and l mt , respectively. If we knew the values of these shadow prices, then solving Eq. (A1) for the remaining shadow prices a is and a isr and the primal values of x ijst , w ins , and y mt becomes very simple. The first-order conditions of the Lagrangian function with respect to x ijst , w ins , and y mt , respectively, can be rearranged as: ais zarit þ
M X m¼1 0
vijstm ðlmt ktm Þ cijst
8i; s ¼ T ; N ; 0;
t ¼ 1; N ; T
ðA2Þ
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
ais zEins aris zarit þ
8i; s ¼ T 0 ; N ; 0 M X m¼1
ðA3Þ
vijstmðl k Þ c tm ijst mt
8i; s ¼ 1; N ; T
ðA4Þ
aris zEins
ðA5Þ
8i; s ¼ 1; N ; T
bt1 Dmt ðymt Þ ¼ mmt
8mt
ðA6Þ
where k tm is applied to a mill that takes wood from areas where the mill is not allowed to harvest or k tm =0 if igI m . If k tm and l mt are known, then Eqs. (A3)– (A5) can be easily solved using dynamic programming because the equations form a recursive system. Once Eqs. (A3)–(A5) are solved,Pthen Eqs. P (6) and (7) ¼ can be usedPto calculate z m mt iaI jaJim vijstm xijst PJi Ptz I and y¯mt ¼ j¼1 i¼1 s¼T 0 vijstm xijst . The variable z mt is the amount of wood going to mill m in period t that is harvested from locations that are restricted to mill m, and y¯ mt is the amount of wood supplied to mill m in period t given the k mt and l mt values. According to the constraint (Eq. (6)), z mt should be less than zero. If z mt N0, then the constraint is violated and the shadow price k mt should be adjusted upward. The supply variable y¯ mt is compared to demand variable y˜ mt that solves b t1D mt (y˜ mt )=l mt . If the deviation d mt =y¯ mt y˜ mt is positive, then mill m is oversupplied given l mt and l mt should be adjusted downward. If d mt is negative, then mill m is undersupplied given that l mt and l mt should be increased. Simple price adjustment heuristics are used to adjust the shadow prices as discussed in Hoganson and Rose (1984) and Hauer and Hoganson (1996). Prices are adjusted and Eqs. (A3)–(A5) are resolved in an iterative manner until the z mt and the d mt values are within a small tolerance of zero. The algorithm is summarized in Appendix B.
Appendix B. The algorithm The solution of the model as described by Hoganson and Rose (1984) proceeds with the following steps: 1.
Use prior information about the problem to estimate the marginal costs of production for
623
each output [lumber, OSB, and chips] and period (i.e., the l mt values). 2. Set k mt =0 for all m and t. 3. Calculate the net dual price for final products (l mt k mt ) for each mill and time period t. 4. Assume that the net dual prices (l mt k mt ) are correct and solve Eqs. (A3)–(A5), using dynamic programming, for the remaining dual variables (a is and a isr values). This finds the maximum land value for each stand type. 5. Determine the primal solution for the x ijst and w ijs values that corresponds to the dual solution in step (4). This part of the primal solution will be feasible with respect to Eqs. (2) and (3). 6. Determine the supply levels y¯ mt , demand levels y˜ mt , and restricted P harvest P levels z mt using PEq. P(6) P zmt ¼ iaI m jaJim vijstm xijst , tz t 1 i y¯mt ¼ Ii¼1 Jj¼1 v x D mt s¼T 0 ijstm ijst and b (y˜ mt )=l mt , respectively. Calculate deviations d mt =y¯ mt y˜ m . 7. Test for primal feasibility. If the d mt and z mt values are small, then stop; the primal solution is both an optimal and a near-feasible solution. Otherwise: 8. Use the d mt and z mt values and a basic understanding of the simple economic relationship between prices and quantities supplied and demanded to reestimate the l mt and k mt dual prices. The dual prices were reestimated using procedures described by Hoganson and Rose (1984) and modified by Hauer and Hoganson (1996). Return to step (3).
References Alavalapati, J.R.R., Luckert, M.K., 1997. Modeling the effect of institutional constraints on short-run timber supply on public land: a case study of quota holders in Alberta. Nat. Resour. Modeling 10 (4), 263 – 282. Cumming, S.G., Armstrong, G.W., 1999. Divided land bases and overlapping tenures in Alberta’s mixedwood forests: a simulation study of policy alternatives. Sustainable Forest Management Network. Working Paper 1999-3. 40 pp. Hauer, G.K., Hoganson, H.M., 1996. Tailoring a decomposition method to a large forest management scheduling problem in northern Ontario. INFOR 34 (3), 209 – 231.
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Hoganson, H.M., Rose, D.W., 1984. A simulation approach to optimal timber management scheduling. For. Sci. 30 (1), 220 – 238. Johnson, K.L., Scheurman, H.L., 1977. Techniques for prescribing optimal timber harvest and investment under different objectives—discussion and synthesis. For. Sci. Monogr. 18 (31 pp.).
Luckert, M.K., 1991. Tenures for mixed wood management: a framework for policy analysis. Forest Economics and Policy Analysis Research Unit. University of British Columbia. Working Paper No. 164, 44 pp.
Estimating the costs of overlapping tenure constraints: a case study in Northern Alberta, Canada David M. Nananga,*, Grant K. Hauerb a
Natural Resources Canada, Canadian Forest Service, 580 Booth Street, Ottawa, Ontario, Canada K1A 0E4 b Department of Rural Economy, University of Alberta, Edmonton, Alberta, Canada T6J 0N6 Received 18 March 2004; received in revised form 1 November 2004; accepted 10 November 2004
Abstract The main objectives of this study were to: (i) incorporate the types of constraints implied by overlapping tenures into a timber supply model; and (ii) estimate the costs associated with various constraints implied by overlapping tenures in Alberta, Canada. This was achieved through a linear programming formulation of the timber supply problem over a 100-year planning horizon, which resulted in more than 5 million decision variables and approximately 118,000 constraints. Given the size of the model, a dual-decomposition procedure to solving large-scale linear programming problems was applied. The results showed that, in general, constraints imposed by overlapping tenures led to inefficiencies in wood allocation and substantial increases in the marginal costs of production. Secondly, the effect of the overlapping tenure constraints was unevenly distributed among mills. The costs to individual tenure holders are highly dependent on how far the mills are from their allowed harvest locations and how far the constraints shift mill harvest areas away from their optimal wood procurement zones. Removal of the constraints leads to a 7% increase in the net present value of the forest. For mills that are located within short distances of their allowed harvest locations, removal of constraints do not significantly lower marginal costs. The results show that overlapping tenure constraints are inefficient and therefore should be removed and better ways of allocating land for harvest should be sought. Although such policies would be efficient, tenure holders who derive an economic advantage from existing arrangements will oppose them. In these cases, some means of compensating mills that lose as a result of more efficient wood allocation may have to be arranged. D 2004 Elsevier B.V. All rights reserved. Keywords: Decomposition techniques; Forest management scheduling; Linear programming; Overlapping tenures; Shadow prices
1. Introduction
* Corresponding author. Tel.: +1 613 947 9023; fax: +1 613 947 9020. E-mail address: [email protected] (D.M. Nanang). 1389-9341/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.forpol.2004.11.003
Changing technology in wood processing in Alberta, Canada has made it possible to utilize species of wood, which previously had no commercial value. The emergence of these species as valuable resources
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
occurred while traditionally valued species often growing as part of the same forest stand also increased in value (Luckert, 1991). This situation led to new management problems associated with the joint production of multiple outputs from forestland. The Alberta Forest Service has been allocating harvesting rights to more than one firm on the same piece of land as one way of ensuring that the newly valued resources are utilized (Luckert, 1991). As a result, overlapping tenures or divided land bases have emerged in Alberta as a significant issue as the number of forest management agreement (FMA) areas increased. The usual configuration of overlapping tenures in Alberta is that one firm has an area-based tenure with rights to harvest over the whole area of the forest as well as rights to harvest all or most of the tree species on the areas. These rights carry obligations for the firm to regenerate harvested areas; plan the sequencing of harvests; and ensure that harvesting operations are sustainable. The full set of rights and obligations is set out in an FMA between the timber firm and the provincial government. The term boverlapping tenureQ comes about when, within the FMA areas, there are embedded volume-based tenures, which are held by other firms. Although the volume-based tenures or quotas are not barea-basedQ like FMAs, quota holder harvests are usually spatially restricted to specific areas within FMA boundaries. In addition, volumebased tenures are usually species-specific. Only certain species may be harvested by the quota holders. To complicate matters further, forestland is usually classified by predominant species and quota holder harvesting rights are usually restricted to the part of the land base where the species specified by the quota predominates. For example, if the volume quota is for conifer, the quota holder may be restricted to harvesting off the part of the land base classified as conifer. In some cases, quota holders may also have rights to incidental volumes harvested off other land bases. Overlapping tenure constraints also interact with other regulations. First, regeneration standards are specified to return forest stands to approximately the same species composition that existed before harvest. Hence, conifer stands are regenerated to return to conifer, deciduous to deciduous, and mixed to mixed stands. This may in some cases prevent stands from
611
being regenerated most cost-effectively from a social perspective. Second, there are often implicit sustainability constraints such as annual allowable cut (AAC) constraints (or even-flow constraints) applied to the land bases, embedded within the larger FMA land base, from which quota holders draw their wood supply. Finally, it is usually unclear as to who has rights to increased allowable cuts that may be obtained by increasing silvicultural input into the forest (the so-called allowable cut effect). A few researchers have previously examined overlapping tenures in Alberta. Cumming and Armstrong (1999) used a simulation approach to conduct a quantitative examination of the costs of overlapping tenures and divided land bases in Alberta. They concluded that the costs of overlapping tenures and divided land bases are substantial enough to justify a thorough examination of forest policy regarding overlapping tenures in Alberta. One other study by Alavalapati and Luckert (1997) modeled the short-run timber supply of quota holders in Alberta in the face of institutional constraints (allowable cut and mill capacity) and fixed stumpage prices using dynamic optimization techniques. The shadow prices of mill processing capacity and allowable cut restrictions were estimated for large, medium, and small tenure holders to reflect the different cost structures of different-sized firms. The results indicated that all categories of quota holders studied incurred substantial costs due to these two institutional constraints, and that simultaneous elimination of both constraints leads to more cost reduction than the combined savings from eliminating each constraint individually. The focus of that study was on quota holders, and therefore did not address the special problems related to overlapping tenures involving FMA holders that are investigated in this study. Despite the apparent inefficiencies in the application of overlapping tenures, the situation persists because quotas are largely historical and politically entrenched rights to species that, in many cases, were granted before the current FMAs were signed. The rights exist as a volume allotment in areas where the desired species exist. Therefore, a key problem is that there is a lot of inertia to reallocate, as many of these small operators have roots in the locations where they operate.
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D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
The objective of this paper is to estimate the costs associated with various constraints implied by overlapping tenures for a case study area in Alberta. Our approach is to formulate a mathematical program-
ming problem for the situation and estimate the costs by incorporating constraints and then relaxing them to represent policy change. Incorporating the constraints implied by overlapping tenures complicates
Table 1 Summary of mill target demands and overlapping tenure constraints Model run
Demand (Eq. (7))
Restrictions on areas that mills are allowed to harvest (Eq. (6))
Demand sites
Demand location
Max mill price ($/m3)
Max mill demanda
Wood type
Allowed locations
Allowed stand types
B A C B A C B
85 85 85 65 65 60 85
220 20 20 215 200 100 220
SW SW SW HW HW SW SW
All
All
Scenario 1
Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1
SW, HW
Scenario 1a
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1
A C B A C B
85 85 65 65 60 85
20 20 215 200 100 220
SW SW HW HW SW SW
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6l Sawmill 1
A C B A C B A C B A C B
85 85 65 65 60 85 85 85 65 65 60 85
20 20 215 200 100 220 20 20 215 200 100 230
SW SW HW HW SW SW SW SW HW HW SW SW
South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1 and 2 North FMA subregion 3 All All North FMA subregion 3 South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1, 2 North FMA subregions 3 All All North subregion 3 All
Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6 Sawmill 1 Sawmill 2 Sawmill 3 OSB mill 4 OSB mill 5 Pulpmill 6
A C B A C B A C B A C
85 85 65 65 60 85 85 85 65 65 60
22 22 240 220 110 230 22 22 240 220 110
SW SW HW HW SW SW SW SW HW HW SW
Baserun
Scenario 1b
Scenario 2
Scenario 3
SW=conifer; HW=deciduous. Subregions refer to FMUs within FMAs. Prices and costs are in 2003 Canadian dollars. a Maximum volume demand per year for the mill in cubic meters.
South FMA and north FMA subregions 1, 2, and 3 North FMA subregions 1, 2 North FMA subregions 3 All All North subregion 3 All
SW SW HW, SW HW, SW SW All
SW, HW SW SW HW, SW HW, SW SW SW, HW SW SW HW, SW HW, SW SW All
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
model formulations in several ways. First, to properly represent the overlapping tenure situation, more than one demand location must be represented so that the different tenure holders may be modeled. To fully capture the costs of constraints, which are spatial both because they are applied over forest management units (FMUs) within an FMA, and because of the different demand locations, multiple supply locations must be modeled. The policy to allow overlapping tenures, which seek to guarantee a supply of timber to quota holders over a long period of time, makes it necessary to test the sustainability and cost of the existing wood supply configuration over a long period of time. Furthermore, since overlapping tenures impose restrictions on where and on what kind of land class wood can be taken, models must keep track of which land classes and which locations desired volumes are being taken from. This is because sawmills and pulpmills use only conifer species, while OSB mills use deciduous species only (Table 1). These temporal and spatial requirements considerably increase the number of decision variables as well as constraints to be modeled. The timber supply model is formulated for two FMA areas in northern Alberta, which have significant overlapping tenure situation. While the description of the land base and the overall overlapping tenure situation are realistic, the model implementation is modified enough to be less realistic in terms of the magnitude of the direction of changes, the wood demands represented, and the number of mills represented. The overlapping tenure constraints that are explicitly examined in this paper are harvest location and predominant species land class restrictions for quota holders. 1.1. Model formulation Two FMA areas in Alberta with a total productive forest area of about 550,000 ha were used in this study. There is one FMA area in the north and one in the south (north and south FMA, respectively) and the spatial units of analyses were 2500-ha grid cells (see Fig. 1). Mathematically, the timber supply problem can be described by the set of equations given in Eqs. (2– 7). The formulation is an extension of the Model II formulation given in Johnson and Scheurman (1977).
613
Fig. 1. The study area showing the north and south FMA areas and the three demand locations.
The objective function (Eq. (1)) maximizes the net benefit of timber products from all mills (firms) with rights to timber on the two FMAs subject to the standard Model II age class constraints, demand constraints, and overlapping tenure constraints. The benefits are the net returns of the value of wood products plus the value of ending forest inventory minus costs of regeneration, harvesting, and transportation. The first term in the objective function (Eq. (1)) is the discounted value of wood products. While our model is general enough to handle downwardsloping demand curves, simplified demand curves are used here, which are perfectly elastic (single price) up to a maximum mill capacity. The second term in Eq. (1) is the value of the ending inventory, whilst the last term is the costs of regeneration, harvesting, and transportation: M X T X
max
t1
"Z
b
m¼1 t¼1
þ
Ji X I T X X
#
ymt
Dmt ð X ÞdX
0
Eins wins
i¼1 j¼1 t¼1
Ji I tz X T X X X i¼1 j¼1 s¼T 0 t¼1
cijst xijst
ð1Þ
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D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
Ji X T X
xijst þ
j¼1 t¼1
Ji X
8i; s ¼ T 0 ; N ; 0
wijs ¼ Ais
j¼1
Ji T X X
xijth þ
j¼1 h¼tþz
Ji X
wijt
j¼1
Ji tz X X
ð2Þ xijst ¼ 0
j¼1 s¼T 0
8i; t ¼ 1; N ; T
ð3Þ
land base regeneration restrictions: ! ! T J T X X X X c xijst þ wijs zd xijst þ wijs jaJic
t¼sþz c
cd
8iaI ; I ; I ; T X
jaJid
t¼sþz
8iaI d
8s ¼ 0; N ; T z !
X
t¼sþz
j¼1 dc
xijst þ wijs zd
d
ð4Þ
J X
T X
j¼1
t¼sþz
! xijst þ wijs
8s ¼ 0; N ; T z
ð5Þ
allowable harvest area and land base restrictions (LBRs): XX vijstm xijst V0 8s; t; and m ð6Þ iaI m jaJim
market clearing restrictions: Ji I tz X X X
vijstm xijst Vymt
8t; m ¼ 1; N ; M
ð7Þ
i¼1 j¼1 s¼T 0
xijst z0
8ijst
wijs z0
8ijs
ymt z0
8mt
Area of stand type i managed with regeneration prescription j, in period s and left as ending inventory c ijst The discounted cost per unit area of managing stand type i with regeneration and market shipping plan j, starting in period s and final harvest in period t x ijst Area managed on stand type i with regeneration prescription and market shipping plan j, starting in period s and final harvest in period t v ijstm The volume per unit of wood products from mill m, in period t, when stand type i is regenerated in period s and managed with prescription and market shipping plan j y mt Quantity of wood demanded by mill m in period t dc Percentage of analysis area that must be regenerated to conifer species dd Percentage of analysis area that must be regenerated to deciduous species Ji The set of regeneration prescriptions and transport destinations for analysis area i; it should be noted that wood from any stand i can be sent to more than one destination Jci Subset of J i that includes regeneration prescriptions that meet the conifer standards Jdi Subset of J i that includes regeneration prescriptions that meet the deciduous standards I c, I d, I dc, I cd Set of conifer, deciduous, mixed deciduous/conifer, and mixed conifer/deciduous land bases, respectively I The number of stand types Im The set of locations or forest types that are not available to mill m Jmi The set of transport/prescriptions that are defined for mill m z Minimum time between regeneration and harvest T The number of planning periods in the planning horizon b Discount factor (using 5% discount rate). w ijs
subject to Model II forest dynamics constraints:
where: D mt (X) The inverse demand function for wood from mill m in period t A is The number of area unit of stand type i in the first period that were regenerated in period s E ijs The discounted value per unit area of managing stand type i with regeneration prescription j, starting in period s and leaving the stand type as ending inventory
Eqs. (2) and (3) are standard Model II age class dynamics constraints (see Johnson and Scheurman, 1977). These equations ensure that total area harvested does not exceed the initial size of the land base and that all areas harvested beyond the first cut were harvested and regenerated previously. Current provin-
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
cial regulations require that conifer land bases be regenerated to predominantly conifer and deciduous land bases to predominantly deciduous species. Constraints (4) and (5) represent these additional regulations. Eq. (6) represents restrictions on where quota holders are allowed to harvest. These constraints can incorporate both spatial restrictions that confine harvest to specific subareas within the FMA area (harvest location restrictions (HLRs)) and restrictions on what type of land base (conifer, mixed conifer, mixed deciduous, and deciduous) may be harvested (LBRs). Eq. (7) implies that volume of wood products produced from all stands managed in the two FMAs should be less than or equal to the mill demands. There are no AAC constraints in this model because AAC is assumed to be highly correlated with mill capacity (maximum mill demand). Increases in mill demand are assumed to drive increases in AAC or, conversely, any increase in AAC is assumed to be absorbed by increases in mill demand or capacity. Hence, in this model, AAC regulations are embedded in the demand structure and are essentially represented by the summation of the mill demand constraints. While Eqs. (2–7) set out the economic problem, the enormous size of the formulation will be difficult to solve, especially if a nonlinear demand system is specified. The model specification for the case study area has 29,885 forest type/location/age class combinations; a 100-year planning horizon with 10 planning periods; a minimum rotation of 40 years; three regeneration prescriptions per stand; a total of 6741 stand types identified by species types, site class, and location; and approximately 10 shipping alternatives for each stand.1 With these assumptions, the resulting model has approximately 5 million decision variables and about 118,000 constraints. Given the difficulty of solving a model of this size, a variant of the dual-decomposition
1
This is based on three species with two size classes for each species, four possible destinations for softwood species, two possible destinations for hardwood species, and the assumption that only half of these alternatives would be available for each stand on average.
615
algorithm proposed by Hoganson and Rose (1984) and later modified by Hauer and Hoganson (1996) was used to solve the problem. Some details of this approach are included in the appendices. 1.2. Model scenarios and overlapping tenure constraints Six hypothetical scenarios were developed to assess the costs of overlapping tenure constraints under different conditions. The scenarios are summarized in Table 1. The Baserun is meant to represent a case where there are no overlapping tenure restrictions. In the Baserun, the model allocates the wood across mills so as to maximize overall net benefits. Scenario 1 is designed to reveal the joint effect of HLRs and LBRs relative to the Baserun, which has no such restrictions. In Scenario 1a, the LBRs are removed, leaving only the HLRs. Scenario 1b drops the HLR and examines the effect of the LBRs only. In Scenario 2, maximum mill demands are assumed to absorb an increase in AAC. The increased AAC is distributed proportionately to all tenure holders, based on their shares of the AAC for each FMA. Scenario 3 uses the same AAC allocations as Scenario 2, but Scenario 3 drops the overlapping tenure constraints. Scenarios 2 and 3 are used to simulate the effect of changes in the AAC and the joint effect of overlapping tenure constraints and changes in AAC on changes in marginal costs of timber production. The second column in Table 1 identifies the demand sites of which there are six. We shall refer to demand locations as A, B, and C. Each demand location has two demand sites, which include sawmills, OSB mills, and pulpmills. Sawmills 1, 2, and 3 are located in demand locations B, A, and C, respectively. OSB Mills 4 and 5 are located at demand locations B and A, respectively. The pulpmill is located at location C. The fourth and fifth columns of Table 1 are sufficient to describe the simplified demand curves for three end products used in this analysis, with demand being perfectly elastic at a maximum price (column 3) and perfectly inelastic along a vertical segment (column 4). The third column shows an estimate of the maximum price in terms of dollars per cubic meter of final product that could be paid at the mill gate. These
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maximum price levels were set based on current estimates of the prices of lumber and OSB. For the pulpmill, the final product is defined as an intermediate product: wood chips. The fourth column shows the maximum volumes that can be consumed by each mill on an annual basis (the right-hand side of Eq. (7)). The maximum volumes for hardwoods and softwoods add up approximately to the total volumes of these species harvested on the north and south FMA areas during the last 5 years. They also add up approximately to the allowable cuts. Hence, wood demands of mills from this area are roughly equivalent to allowable cuts levels. The breakdowns of volumes by mill are based on the percentage shares that each mill is allocated under the existing FMA and quota agreements. The last two columns in Table 1 summarize the overlapping tenure constraints (Eq. (6)). The second to the last column shows the allowed locations from which each mill can harvest (HLR), while the last column shows the cover types from which each mill is allowed to harvest (LBR). As pointed out earlier, these restrictions in harvest locations are a result of the historical development of overlapping tenures and do not necessarily reflect a desire to optimize overall net benefits from timber harvest. Sawmill 1 in location A and the two OSB mills represent the FMA tenure holder’s demand locations. As part of the tenure agreements, the OSB mills may harvest from anywhere in the two FMA areas. However, in Scenarios 1, 1a, and 2, Sawmill 1 is limited to harvest in the south FMA and only parts of the northern FMA, while the other sawmills and the pulpmill may harvest only from specified portions of the northern FMA. In the Baserun and Scenario 3, these constraints are eliminated entirely so that wood may flow to any mill from any location.
2. Results and discussion Net present values for each scenario are presented in Table 2.2 Comparisons of the Baserun and Scenarios 1, 1a, and 1b show that the Baserun has
2 All prices and costs are in 2003 Canadian dollars. A real discount rate of 5% was used.
Table 2 Net present values for the six scenarios Model run
Objective function value (109 $)a
Baserun Scenario Scenario Scenario Scenario Scenario
2.311 2.140 2.142 2.154 2.302 2.477
1 1a 1b 2 3
a Values are in 2003 Canadian dollars, discounted with a 5% real discount rate.
the highest NPV as expected. The total loss in NPV due to the constraints in Scenario 1 as compared to the Baserun is $171 million, which represents a loss of about 7% of the objective value. However, having both constraints in place only marginally increases the cost of the constraints. From Table 2, imposing HLRs alone decreases NPV by $169 million. Imposing LBRs on top of HLRs decreases NPV by only $2 million. Similarly, imposing LBRs alone decreases NPV by $157 million, whilst imposing harvest area restrictions on top of the LBRs only decreases NPV by $14 million. Shadow price estimates on mill demand constraints, which may be interpreted as marginal costs of production, for the Baserun and Scenario 1 reveal a number of important patterns (Figs. 2–4). First, we describe some of the general patterns. Marginal costs for the Baserun tend to increase over the planning horizon for the sawmills, decrease for the OSB mills, and stay flat or increase for the pulpmills. The increased marginal costs observed are due to long-term scarcities that emerge as more of the wood is harvested off the FMAs. Since the Baserun does not contain any overlapping tenure restrictions, the only reason for increasing marginal costs is scarcity of wood over time, forcing firms to harvest from poorer sites. Shadow prices for OSB mills for the Baserun remain the same or slightly decrease over the planning horizon (Fig. 2). This reflects a relative abundance of aspen wood on the two FMAs. Another pattern that can be seen in Fig. 3 is that marginal costs for all sawmills tend to be closer together when overlapping tenure constraints are relaxed than when they are applied. This occurs
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Fig. 4. Comparison of shadow prices for pine and spruce chips for the Baserun and Scenario 1. Fig. 2. Shadow prices for OSB mills for the Baserun and Scenario 1.
Fig. 3. Comparison of the shadow prices for lumber mills for the Baserun and Scenario 1.
because the only major difference in costs in our model that can exist once overlapping tenure constraints are removed is in transport costs to the mills. At the margin of each mill’s woodshed, the value of sending wood to the competing mills will be roughly equivalent, which agrees with the equimarginal principle. Overlapping tenure woodsheds are not determined by economic criteria and hence we see a divergence in marginal costs. We now turn to the effects of overlapping tenure constraints on marginal costs of production. Marginal costs for Sawmill 3 are significantly higher when overlapping tenure constraints are imposed than in the Baserun. For Sawmill 2, marginal costs are higher when overlapping tenure constraints are imposed except in periods 7 and 10, when they are roughly equivalent. However, for Sawmill 1, the marginal costs, while higher in the first four periods, actually decrease below the marginal costs in the last six periods when there are no overlapping tenure constraints. The overlapping tenure constraints restrict some mills’ harvesting to certain locations and land bases. Hence, one expects costs to drop
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once the constraints are removed. The marginal costs for Sawmill 3 are higher with the constraints because the locations that this mill is allowed to harvest from are outside its woodshed if no constraints were imposed. These results show that although dropping the constraints results in an overall increase in net returns and reductions in costs, the gains are not evenly spread across mills and over time. In this
case, for Sawmill 1, marginal costs are actually lower in the long run when there are overlapping tenure constraints. Insights into why the marginal costs change in the direction they do can be gained by examining the wood procurement zones for each mill. The wood procurement zones for Sawmill 1 are shown in Fig. 5a and b. In the Baserun, without overlapping tenure
Fig. 5. Wood procurement zones for Sawmills 1 and 2 for the whole planning horizon.
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
constraints, Sawmill 1 takes wood from locations that are closer to the mill compared to when there are restrictions. When there are restrictions, harvests for Sawmill 1 spread over a larger area and, in particular, they spread into the northern parts of the northern FMA area. Marginal costs increase for Sawmill 1 in the short run because Sawmills 2 and 3 are forced to harvest in areas where it is more profitable for Sawmill 1 to harvest, thus pushing Sawmill 1’s harvest into less profitable areas. However, Sawmill 1’s marginal costs eventually decrease when restrictions are imposed because, in the long run, there is more wood available to Sawmill 1 because Sawmills 2 and 3 are restricted to small areas within the FMA area. The optimal harvest locations (i.e., without HLR and LBR) for Sawmill 3 are shown in Fig. 6a, which is dramatically different from the woodshed with overlapping tenure constraints shown in Fig. 6b. Part of Sawmill 3’s increase in marginal costs will be due to increased transport costs and part will be due to lack of long-term flexibility and scarcity that result from being constrained to harvest only in the areas shown in Fig. 6b. Similar results for Sawmill 2 can be observed by examining Fig. 5c and d. Fig. 4 shows the marginal costs for softwood chips or pulpwood, and Fig. 6c and d shows the wood procurement zones with and without overlapping tenure restrictions. Removal of the overlapping tenure constraints results in a decrease in marginal costs in all planning periods. The woodshed without the constraints shows that wood destined for the pulpmill is harvested from almost every part of the two FMAs. The reason for the differences in the wood procurement zones appears to be related to wood sorting. There are efficiencies built in the model so that stands can be scheduled to deliver product to multiple mill destinations. When there is no constraint on wood harvesting, wood is taken from many places in the management unit because pulpwood components of conifer stands are shipped to the pulpmill and sawlogs are shipped to sawmills (not necessarily to the same location or tenure holder). When the constraints are put in place, the pulpwood that could have been delivered from the north of Demand Centre A (see Fig. 6c) (which would have been efficient) can no longer be delivered under the constraints. All the pulpwoods
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must be delivered from within the narrowly defined supply area of Fig. 6d. 2.1. Marginal costs of overlapping tenure constraints Table 3 shows the shadow prices or marginal costs of the overlapping tenure constraint set given by Eq. (6). All three sets of shadow prices are described by the same constraint except that the set over which the summation takes place is different in each case. The first set is a combination of restrictions on harvest area and land bases where mills can take their wood. The second is the harvest area restrictions alone, whilst the third is the LBRs. These shadow prices represent the marginal reduction in the objective function value as a result of not allowing a small amount of wood to be harvested from the areas (or land base) from which the mill is restricted from harvesting. Hence, since the shadow price for Sawmill 2 is $3.62/m3 in the first period, the overall objective function would increase by $3.62 for every cubic meter of wood that it could harvest from outside the area from which it is currently allowed to harvest. An examination of Table 3 reveals that for the combined harvest area and LBRs, the shadow prices on Sawmill 3 and the pulpmill (both in demand location C) are higher than the other mills. This is because the FMU allocated for harvesting to these two mills is far outside the two mills’ woodshed if there were no restrictions. In contrast, Sawmills 1 and 2 are located close to the FMUs that they have harvest rights to (compare Fig. 5c and d with Fig. 6a and b), and so the marginal costs of the overlapping tenure constraints are lower. The shadow prices on the harvest area restrictions are much larger than those of LBRs. This occurs because the LBRs prohibit a mill from harvesting deciduous land base if it is a conifer mill and vice versa—so mills are being prevented from harvesting wood from areas that they are less likely to want to harvest in the first place. Hence, the harvest area restrictions are more binding than the LBRs. 2.2. Effect of increased AAC on NPV and marginal costs Next, we consider the effect on marginal costs of the three final products following an increase in
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Fig. 6. Wood procurement zones for Sawmill 3 and Pulpmill 6 for the whole planning horizon.
AAC with and without overlapping tenure constraints. Table 4 shows the average 10-year differences in marginal costs with and without overlapping tenure constraints for all six mills. As pointed out earlier, raising the AAC requires investment from the firms over and above what is required by regulation, however, there is no clear mechanism
for distributing the increased AAC among the tenure holders. In this study, we assumed that the increased AAC is distributed proportionately to all tenure holders, based on their shares of the AAC for each FMA. A comparison of the difference in marginal timber costs in Scenarios 2 and 3 to the difference in
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Table 3 Final shadow price estimates of the harvest area and land base restrictions for Scenario 1 by period Constraint type
Planning period 3
4
5
6
7
8
9
10
Harvest location and land base restrictions Sawmill 1 3.62 4.73 Sawmill 2 6.93 6.45 Sawmill 3 6.58 6.92 Pulpmill 6 15.35 13.85
1
2
9.62 8.56 10.27 15.37
9.36 8.88 11.16 18.97
13.15 13.03 17.79 20.78
15.67 16.12 22.89 26.94
21.96 23.23 33.92 39.59
18.69 20.88 35.83 37.96
22.10 23.01 43.73 46.32
22.39 23.32 47.88 47.88
Harvest location restrictions Sawmill 1 3.06 Sawmill 2 6.93 Sawmill 3 6.58 Pulpmill 6 15.35
4.28 5.67 7.20 12.89
9.41 9.44 10.67 15.37
8.49 7.99 10.30 18.14
13.70 13.30 17.16 18.87
16.73 16.58 21.57 24.33
22.30 22.09 29.98 33.38
23.68 23.51 27.10 32.00
26.62 27.10 27.29 38.52
27.13 26.09 32.27 37.15
Land base restrictions Sawmill 1 0.00 Sawmill 2 6.25 Sawmill 3 2.97 Pulpmill 6 7.49
0.00 1.55 1.53 7.34
0.00 1.03 0.00 9.49
0.00 6.75 0.06 11.87
0.00 6.72 1.17 14.15
0.00 1.37 0.16 9.02
0.00 2.81 0.21 11.56
0.00 5.53 0.32 13.23
0.00 5.57 3.42 22.34
0.00 0.59 0.15 10.61
No OSB results are shown because no harvest location or land base restrictions were placed on the OSB mills.
marginal costs for Scenario 1 and the Baserun shows that overlapping tenure constraints have a greater effect when allowable cut levels are higher. With the exception of Sawmill 1 and OSB mill 1, the marginal costs of the demand constraints all increased by a greater degree when overlapping tenure constraints were present (compare rows 1 and 2 in Table 4). Another way of looking at the costs of the overlapping tenure constraints is to examine the change in marginal costs of production when allowable cut is increased with and without the constraints. Results in rows 3 and 4 of Table 4 show that when allowable cut is increased, the marginal cost increases are much lower when there are no overlapping tenure constraints present. Increasing cut levels is less expensive when the constraints are not present.
It is important to point out that changes in the assumptions of the model will affect the conclusions and management implications for tenure holders. For example, we assume a 100-year planning horizon during which the tenure holder abides by all the requirements of the tenure agreement, which is not quite realistic. It is possible that, over time, the tenures could be renegotiated with different allocation of rights that would reduce the costs of the overlapping tenure constraints. For the three products that were modeled, an increase in the number of OSB mills, for example, could lead to more deciduous species being harvested, and hence an increase in the costs of the constraints. Increases in the prices of final products will increase profitability for the firms, although it is unlikely to affect the costs of the constraints imposed by overlapping tenures.
Table 4 Average 10-year increase in marginal costs ($/m3) between selected scenarios Differences Scenario Scenario Scenario Scenario
1 2 2 3
minus minus minus minus
Baserun Scenario 3 Scenario 1 Baserun
Sawmill 1
Sawmill 2
Sawmill 3
OSB mill 1
OSB mill 2
Pine chips
Spruce chips
0.736 3.96 1.32 0.57
0.43 1.71 1.62 0.57
15.87 33.61 18.23 0.05
8.05 7.90 0.11 0.03
7.56 7.61 0.35 0.24
21.16 38.34 20.94 3.69
30.37 45.57 15.90 0.55
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3. Conclusions This study applied an optimization approach to estimate the cost of overlapping tenure constraint on FMA areas in northern Alberta. Overlapping tenures continue as a policy instrument in Alberta because quotas are largely historical and politically entrenched rights to species that, in many cases, were granted before FMAs were signed. The results from the various scenarios investigated showed that overlapping tenure constraints are costly. The costs to individual tenure holders are highly dependent on how far the mills are from their allowed harvest locations and how far the constraints shift mill harvest areas away from their optimal wood procurement zones. Although, in general, removal of the constraints leads to decreased costs, the benefits of removing constraints are unevenly distributed among tenure holders. Removal of the constraints leads to a 7% increase in the net present value of the forest. For mills that are located within short distances of their allowed harvest locations, removal of constraints does not significantly lower marginal costs. Increases in AAC are more costly when overlapping tenure constraints are present than when they are removed. The results here suggest that overlapping tenure constraints should be removed and better ways of allocating land should be sought. One possible policy solution would be for the Alberta Forest Service to grant tenure holders comprehensive rights to all species within a licensed area (Luckert, 1991). Tenure holders could negotiate with each other to trade and/or sublease their rights to certain species that cannot be utilized in their mills or are in locations that are too far away from their mill locations. This would likely push the wood allocation closer to that of the Baserun as wood would tend to find its highest valued use. While the results here show overall net benefit improvement from removing overlapping tenures, they also suggest that, in some cases, removal of overlapping tenure constraints may decrease flexibility for some mills, resulting in increases in costs for those mills. Although policies to remove overlapping tenure constraints would be efficient, tenure holders who derive an economic advantage from existing arrangements will oppose them. In these cases, some means of compensating mills that lose as
a result of more efficient wood allocation may have to be arranged.
Appendix A. The dual-decomposition approach: theory We start with the Lagrangian function for Eqs. (2)– (4) and Eqs. (6)–(7): M X T X
min Lða; ar ; k; lÞ ¼ max r x;y;w
a;a ;k;m
"Z
#
ymt
m¼1 t¼1
Dmt ð X ÞdX þ
0
Ji X I T X X
cijst xijst þ
"
Ais
Ji X T X j¼1 t¼1
Ji X tz X
t¼1 s¼T 0 m¼1
þ
#
wijs þ
Ji T X X
"
lmt
t¼1 m¼1
"
ktm
ais
I T X X
arit
i¼1 t¼1
xijth
j¼1 h¼tþz
T 0 M X X X
T X M X
Ji X j¼1
xijst
j¼1 s¼1
þ
xijst
I 0 X X i¼1 s¼T
i¼1 j¼1 s¼T 0 t¼1
Eins wins
i¼1 j¼1 t¼1
Ji I tz X T X X X
"
bt1
Ji X
#
wijt
j¼1
XX
#
vijstm xijst
iaI m jaJim Ji I tz X X X
#
vijstm xijst ymt
i¼1 j¼1 s¼T 0
ðA1Þ Our solution approach proceeds by relaxing the requirement for strict feasibility of Eqs. (6) and (7) and assuming that we have knowledge of the shadow prices on these constraints, k mt and l mt , respectively. If we knew the values of these shadow prices, then solving Eq. (A1) for the remaining shadow prices a is and a isr and the primal values of x ijst , w ins , and y mt becomes very simple. The first-order conditions of the Lagrangian function with respect to x ijst , w ins , and y mt , respectively, can be rearranged as: ais zarit þ
M X m¼1 0
vijstm ðlmt ktm Þ cijst
8i; s ¼ T ; N ; 0;
t ¼ 1; N ; T
ðA2Þ
D.M. Nanang, G.K. Hauer / Forest Policy and Economics 8 (2006) 610–624
ais zEins aris zarit þ
8i; s ¼ T 0 ; N ; 0 M X m¼1
ðA3Þ
vijstmðl k Þ c tm ijst mt
8i; s ¼ 1; N ; T
ðA4Þ
aris zEins
ðA5Þ
8i; s ¼ 1; N ; T
bt1 Dmt ðymt Þ ¼ mmt
8mt
ðA6Þ
where k tm is applied to a mill that takes wood from areas where the mill is not allowed to harvest or k tm =0 if igI m . If k tm and l mt are known, then Eqs. (A3)– (A5) can be easily solved using dynamic programming because the equations form a recursive system. Once Eqs. (A3)–(A5) are solved,Pthen Eqs. P (6) and (7) ¼ can be usedPto calculate z m mt iaI jaJim vijstm xijst PJi Ptz I and y¯mt ¼ j¼1 i¼1 s¼T 0 vijstm xijst . The variable z mt is the amount of wood going to mill m in period t that is harvested from locations that are restricted to mill m, and y¯ mt is the amount of wood supplied to mill m in period t given the k mt and l mt values. According to the constraint (Eq. (6)), z mt should be less than zero. If z mt N0, then the constraint is violated and the shadow price k mt should be adjusted upward. The supply variable y¯ mt is compared to demand variable y˜ mt that solves b t1D mt (y˜ mt )=l mt . If the deviation d mt =y¯ mt y˜ mt is positive, then mill m is oversupplied given l mt and l mt should be adjusted downward. If d mt is negative, then mill m is undersupplied given that l mt and l mt should be increased. Simple price adjustment heuristics are used to adjust the shadow prices as discussed in Hoganson and Rose (1984) and Hauer and Hoganson (1996). Prices are adjusted and Eqs. (A3)–(A5) are resolved in an iterative manner until the z mt and the d mt values are within a small tolerance of zero. The algorithm is summarized in Appendix B.
Appendix B. The algorithm The solution of the model as described by Hoganson and Rose (1984) proceeds with the following steps: 1.
Use prior information about the problem to estimate the marginal costs of production for
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each output [lumber, OSB, and chips] and period (i.e., the l mt values). 2. Set k mt =0 for all m and t. 3. Calculate the net dual price for final products (l mt k mt ) for each mill and time period t. 4. Assume that the net dual prices (l mt k mt ) are correct and solve Eqs. (A3)–(A5), using dynamic programming, for the remaining dual variables (a is and a isr values). This finds the maximum land value for each stand type. 5. Determine the primal solution for the x ijst and w ijs values that corresponds to the dual solution in step (4). This part of the primal solution will be feasible with respect to Eqs. (2) and (3). 6. Determine the supply levels y¯ mt , demand levels y˜ mt , and restricted P harvest P levels z mt using PEq. P(6) P zmt ¼ iaI m jaJim vijstm xijst , tz t 1 i y¯mt ¼ Ii¼1 Jj¼1 v x D mt s¼T 0 ijstm ijst and b (y˜ mt )=l mt , respectively. Calculate deviations d mt =y¯ mt y˜ m . 7. Test for primal feasibility. If the d mt and z mt values are small, then stop; the primal solution is both an optimal and a near-feasible solution. Otherwise: 8. Use the d mt and z mt values and a basic understanding of the simple economic relationship between prices and quantities supplied and demanded to reestimate the l mt and k mt dual prices. The dual prices were reestimated using procedures described by Hoganson and Rose (1984) and modified by Hauer and Hoganson (1996). Return to step (3).
References Alavalapati, J.R.R., Luckert, M.K., 1997. Modeling the effect of institutional constraints on short-run timber supply on public land: a case study of quota holders in Alberta. Nat. Resour. Modeling 10 (4), 263 – 282. Cumming, S.G., Armstrong, G.W., 1999. Divided land bases and overlapping tenures in Alberta’s mixedwood forests: a simulation study of policy alternatives. Sustainable Forest Management Network. Working Paper 1999-3. 40 pp. Hauer, G.K., Hoganson, H.M., 1996. Tailoring a decomposition method to a large forest management scheduling problem in northern Ontario. INFOR 34 (3), 209 – 231.
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Hoganson, H.M., Rose, D.W., 1984. A simulation approach to optimal timber management scheduling. For. Sci. 30 (1), 220 – 238. Johnson, K.L., Scheurman, H.L., 1977. Techniques for prescribing optimal timber harvest and investment under different objectives—discussion and synthesis. For. Sci. Monogr. 18 (31 pp.).
Luckert, M.K., 1991. Tenures for mixed wood management: a framework for policy analysis. Forest Economics and Policy Analysis Research Unit. University of British Columbia. Working Paper No. 164, 44 pp.