Forest harvesting to optimize timber production and water runoff1

PII:

Socio-Econ. Plann. Sci. Vol. 32, No. 4, pp. 277±293, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-0121/98 $19.00 + 0.00 S0038-0121(97)00038-4

Forest Harvesting to Optimize Timber Production and Water Runo€{ JOHN ROWSE{ Department of Economics, The University of Calgary, Calgary, Alberta T2N 1N4, Canada

and CALUM J. CENTER Zi€ Energy Group, 1117 Macleod Trail SE, Calgary, Alberta T2G 2M8, Canada AbstractÐTimber and water runo€ are joint forest products, and augmentation of runo€ can occur when timber harvest uses smaller but costlier cut blocks. If runo€ from a forested catchment area drains into a water-scarce region, then this interesting question arises: how best can harvesting in the catchment area be tailored to increase runo€ to the water-scarce region? To examine the economics of joint production for a forested area in Canada, a linear programming model maximizing net present value of timber production and water runo€ is run for three block sizes. Including a moderate or high value for water runo€ leaves the optimal harvest pattern unchanged and does not cover the additional costs of smaller blocks. Yet, smaller blocks yield other bene®ts to society which, if valued monetarily, might make them preferred. To realize the bene®ts of smaller blocks would likely require public sector involvement for devising institutions/contracts linking the user of increased runo€ with the harvester whose practices are modi®ed. # 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION In multiple-use analyses of forested lands, the nexus between timber harvesting methods and water runo€ appears to have received scant attention. Yet it is known that, in areas characterized by snow-dominated hydrologic processes, harvest cut blocks of certain sizes and shapes tend to catch more snow, which is later released as increased water runo€. Suppose that runo€ from a forested water catchment area drains into a water-scarce region. The area±region pair might be neighbouring states or provinces within a country, or neighbouring countries. An interesting question then arises: how best can harvest practices in the catchment area be tailored to provide increased water runo€ to the water-scarce region? Because timber harvesting practices have rarely (if ever) been modi®ed to increase catchmentbasin water runo€, a plausible conjecture is that the economies from such modi®cations are small or nonexistent. Yet, even addressing this question analytically is complicated. First, can institutions evolve to link the user of increased runo€ with the harvester whose practices are modi®ed? If not, then contractual agreements to modify harvest practices cannot easily be reached. Where neighbouring political jurisdictions are involved, public sector participation will likely prove critical for devising the necessary institutional/contractual mechanisms. Second, even if practices are altered, increased runo€ usually occurs in the spring, when its value is relatively small. Thus, storage facilities may be needed to allow the best use of increased runo€. If no storage facilities exist, then their construction and best use over time must be planned. Hence, a related dynamic optimization problem, perhaps involving ¯ood control and recreational water use, must be solved as well. Finally, the value of incremental runo€ must be found by compar-

{This paper is based upon research reported more fully in Center [11]. We extend our sincere thanks to Richard Moll of Statistics Canada for his help with this work and to Richard, Dianne Draper and Elizabeth Wilman for helpful comments. We are responsible for any errors that remain. {Author for correspondence.

277

278

John Rowse and Calum J. Center

ing optimal forest use taking no account of runo€ with optimal forest use taking account of runo€, thus solving two complicated problems. In analyzing the economics of modifying forest harvesting to increase water runo€, we simplify considerably. We ignore institutional arrangements and storage and focus on the possible incremental value from increasing runo€. We utilize linear programming and examine three di€erent cut block methods. An uncommon feature of our work is that we model saw wood and pulp wood rather than wood ®ber alone. We solve our model without and with harvest smoothing constraints, which restrict harvest adjustments over time. Our analysis area is a portion of the Bow-Crow forest on the eastern slopes of the Rocky Mountains in Alberta, Canada, which drains into the North and South Saskatchewan Rivers. Previewing our results, we ®nd that under di€erent assumptions about the value of water runo€ and plausible higher costs of smaller harvest cut blocks, the total value of additional runo€ does not compensate for the increased costs of smaller blocks. The value of incremental runo€ stands at only 1±2% of net forest bene®ts in each case. However, even though our measured bene®ts of modifying harvest practices do not outweigh the costs of smaller cut blocks, there exist other bene®ts of smaller blocksÐa larger road system (promoting better response to forest ®res), improved wildlife habitat, and increased amenity valuesÐthat could argue for adopting smaller blocks, at least from a societal perspective. This paper is organized as follows. The model is presented next, after which the base and comparative cases are set forth. Model ®ndings are presented and subsequently interpreted, and the sensitivity analyses then discussed. Concluding remarks are o€ered in the ®nal section. THE MODEL Preliminary remarks Linear programming (LP) forestry model formulations are suciently well known that they are typed according to Models I and II; see, for example, Reed [21]. We utilize a variant of agestructured Model II from Reed and Errico [22], which Moll [17] dubs Model III, and we represent joint timber/water production according to the harvest cut block method. For simplicity, our approach is deterministic rather than stochastic.{ Government harvest regulations require that cut block adjacency conditions be met. We allow for adjacency and road construction by making these elements exogenous. To do otherwise could lead to a prohibitively large model. Among others, Navon [19], Weintraub and Navon [27] and Gunn [13, 14] argue that a road network for a given cut block strategy can be speci®ed exogenously. We represent adjacency by insisting that only a particular fraction of the total area can be harvested during each period.{ The model in detail Because LP forestry models have long been utilized, we present our model concisely and concentrate on modi®cations unique to our work. Moreover, in formulating our model we occasionally borrow from the GAMS modelling language (see Brooke et al. [9]), which we used in our work. Our model resembles those of Reed and Errico [23] and Moll and Chinneck [18], but there are some noteworthy di€erences. Sets and parameters Sets specify the domains of variables and parameters, and are de®ned for age classes (A), time periods (T), and cut block methods (J). In Table 1 (and others), we name a model element, de®ne it and list its units. To simplify exposition, we occasionally let a set name (e.g. T) denote a set index. Tree ages are represented by decade. Index A10 represents trees of seedling age to age 10, A20 represents trees of age 10 to age 20, and so on; A150 represents trees older than 140 years. {We brie¯y discuss the requirements of a stochastic approach below, in our section on sensitivity analysis. {The precise locations of the actual cut blocks are not speci®ed and are left to be determined at the operational stage of planning.

Forest harvesting to optimize timber production and water runo€

279

Table 1. Model sets and parameters Sets {A10, A20, . . . A150} {T1, T2, . . ., T25} {M1, M2, M3} Parameters

A T J

Age classes Time periods Cut block methods

PSAW PPULP VWAT STUMP R GAMMA M

Sales price of saw wood Sales price of pulp wood Average value of water Provincial government harvest stumpage fee Annual discount rate Maximum/minimum allowed percentage change in volume between periods Coecient transforming the per-period value of water runo€ into the present valued sales of water runo€ for ten consecutive years Total hectarage of analysis area TOTAREA = aA S0(A), where S0(A) is de®ned below. Maximum proportion of TOTAREA that can be harvested in each period Initial coniferous mixed wood timber inventory Water runo€ associated with harvesting using cut block J Gross merchantable volume of wood ®ber of age class A Net saw wood volume of age class A Discount factor DELT(T) = (1 + R)ÿ10(T ÿ 1) Infrastructure cost of building the road network associated with cut block method J Average harvesting cost, excluding stumpage fees, age classes A50 and older Average harvesting cost, including stumpage fees, age classes A50 and older

TOTAREA Z S0(A) V(J) GMV(A) NSV(A) DELT(T) C(J) MC1(J,A) MC2(J,A)

(decades) (decades) (hectaresÐha) (109$ per 106 (109$ per 106 (109$ per 106 (109$ per 106 (%) (%) (no units)

m3) m3) m3) m3)

(103 ha) (no units) (103 ha) (106 m3 per 1000 ha) (106 m3 per 103 ha) (106 m3 per 103 ha) (no units) (109 $) (109 $ per 106 m3) (109 $ per 106 m3)

Table 2. Model variables and objective functions X(T,A) H(T,A) PULP(T) SAW(T) AVOL(T) PHI1 PHI2 PHI3 PHI4

Decision variables Area of age class A timber harvestable in period T Area of age class A timber harvested in period T Supply of pulp wood in period T Supply of saw wood in period T Water runo€ in period T Bene®t without stumpage fees and without water constraints Bene®t with stumpage fees present but without water constraints Bene®t without stumpage fees but with water constraints present Bene®t with stumpage fees and water constraints present

(103 (103 (106 (106 (106 (109 (109 (109 (109

ha) ha) m3 per period) m3 per period) m3) $) $) $) $)

Bene®t PHI di€ers by model run and is the present-valued di€erence between (i) sales revenues of saw wood, pulp wood and water, and (ii) costs of timber harvesting and infrastructure. Objective function The objective function to be maximized is the net present value (in 109 $) of harvested saw wood, pulp wood and water runo€. For a model run, it is either PHI1, PHI2, PHI3 or PHI4. PHI1 = aT (DELT(T) * (PSAW * SAW(T) + PPULP * PULP(T))) ÿ aT aA150 A = A50 aJ DELT(T) * MC1(J,A) * GMV(A) * H(T,A) ÿ aJ C(J) A150 PHI2 = aT (DELT(T) * (PSAW * SAW(T) + PPULP * PULP(T))) ÿ aT aA = A50 aJ DELT(T) * MC2(J,A) * GMV(A) * H(T,A) ÿ aJ C(J) PHI3 = aT (DELT(T) * (PSAW * SAW(T) + PPULP * PULP(T) + M * VWAT * AVOL (T))) ÿ aT aA150 A = A50 aJ DELT(T) * MC1(J,A) * GMV(A) * H(T,A) ÿ aJ C(J) A150 PHI4 = aT (DELT(T) * (PSAW * SAW(T) + PPULP * PULP(T) + M * VWAT * AVOL(T))) ÿ aT aA = A50 aJ DELT(T) * MC2(J,A) * GMV(A) * H(T,A) ÿ aJ C(J)

Hence, the age-class set A = {A10, A20, . . . , A150}. We utilize 25 10-year time periods, yielding a time frame of 2.5 centuries, and thus the time-period set T = {T1, T2, . . ., T25}.{ Set J = {M1, M2, M3} indexes the three cut block methods we consider. Model data are speci®ed as parameters and these are also de®ned in Table 1. Most parameter names and de®nitions are straightforward, but we discuss some of them below. Table 4 lists most parameter values. Decision variables and the objective function Principal variables are X(T,A), the area of age class A timber harvestable in period T, and H(T,A), the area harvested (see Table 2). PULP(T) and SAW(T), pulp wood and saw wood production, and AVOL(T), water runo€, measure forestry outputs. PHI1, PHI2, PHI3 and {The length of each time period and the number of periods are important model choices. For instance, Moll [17] and Reed and Errico [22] use models with 20-year periods. Since using 20-year periods loses detail in establishing a rotation age, and we prefer a 10-year period.

280

John Rowse and Calum J. Center Table 3. Model constraints

STBAL1(T + 1,A) STBAL2(T + 1,A) STBAL3(T + 1,A) TOTHA(T) CUTX(T,A) BALANCE(T) SBAL(T) S1(T) S2(T) SP1(T) SP2(T) CAVOL(T)

Stock balance equation for harvested hectares X(T + 1,A10)= aA H(T,A) Stock balance equation for middle age-classes X(T + 1,A) = X(T,A ÿ 1) ÿ H(T,A ÿ 1) Stock balance equation for oldest age-class X(T + 1,A) = (X(T,A ÿ 1) ÿ H(T,A ÿ 1) + (X(T,A) ÿ H(T,A)) Maximum number of hectares harvestable in period T aA H(T,A) RZ * TOTAREA Area cut restriction H(T,A) R X(T,A) Merchantable volume PULP(T) + SAW(T) RaA GMV(A) * H(T,A) Sawlog volume SAW(T)R aA NSV(A) * H(T,A) Maximum allowed decline in saw wood production SAW(T) r(1 ÿ GAMMA) * SAW(T ÿ 1) Maximum allowed increase in saw wood production SAW(T)R (1 + GAMMA) * SAW (T ÿ 1) Maximum allowed decline in pulp wood production PULP(T) r(1 ÿ GAMMA) * PULP(T ÿ 1) Maximum allowed increase in pulp wood production PULP(T) R(1 + GAMMA) * PULP(T ÿ 1) Water runo€ using cut block method J AVOL(T) = V(J) * aA H(T,A)

(103 ha) T = {T1, T2, . . ., T24} (103 ha) T = {T1, T2, . . ., T24} and A = {A20, . . . , A150} (103 ha) A = A25 and T = {T1, . . ., T24} (103 ha) 8T (103 ha) 8T,A (106 m3 per period) 8T (106 m3 per period) 8T (106 m3 per period) T = {T2, . . . , T25} (106 m3 per period) T = {T2, . . . , T25} (106 m3 per period) T = {T2, . . . , T25} (106 m3 per period) T = {T2, . . . , T25} (106 m3 per period) 8T

PHI4 are four objective functions; each is the di€erence between present-valued sales revenues and harvest costs. PHI1 is the net bene®t when stumpage fees are not represented and water runo€ is not valued; PHI2 includes stumpage fees but does not value water runo€; PHI3 does not represent stumpage fees but values water runo€; and PHI4 includes stumpage fees and values water runo€. Harvest area restrictions Harvest area restrictions, consisting of timber production equations and harvest constraints, form the ®rst ®ve classes of constraints in Table 3. Following Moll and Chinneck [18], we represent three stages of timber growth. Equations STBAL1, representing stage 1, de®ne the stock of hectares in the youngest age class of each period as the sum of all hectares harvested in the previous period. STBAL2 represent the ageing of unharvested hectares: the amount not harvested in an age class adds to the stock of the next older age class in the following period. Finally, STBAL3 specify that unharvested hectares of the oldest age class add to the total hectarage of that class. The adjacency rule states that if a stand is harvested, a certain number of stands comprising an area surrounding the cut block must not be harvested for a speci®ed amount of time. In constraints TOTHA(T), we represent adjacency by allowing only a fraction Z of the entire analysis area to be harvested in each period. Finally, constraints CUTX(T,A) ensure that age class A hectares harvested never exceed the harvestable stock. Constraints on merchantable volumes We distinguish between saw wood and pulp wood as timber products because di€erent markets exist for these products and their volumes change di€erently as timber ages. Gross merchantable volume coecients GMV(A) indicate how pulp volume changes as trees age, while net saw wood volume coecients NSV(A) indicate how saw volume changes. PULP(T) and SAW(T) are limited by the pulp volume in the BALANCE(T) constraints of Table 3, while constraints SBAL(T) limit saw wood production using the NSV(A) coecients. Together, BALANCE(T) and SBAL(T) allow all timber to be harvested as pulp wood, but only a fraction to be harvested as saw wood. Policy constraints Common policy constraints in forestry models restrict timber production at any time as well as how it may change over time. Pearse [20] justi®es such constraints on the basis of national security, multiple use, moral obligation, and economic stability. Harvest smoothing constraints

Forest harvesting to optimize timber production and water runo€

281

Table 4. Model price, cost, yield and stock data Product prices Product Saw wood Pulp wood Water

Establishment costs and water yields

Price ($/m3)

Harvest method J

200 45 0.08

M1 M2 M3

Age-invariant harvest costs

Cost Item Log hauling Stumpage Reforestation Holding and protection Road maintenance: High (5 ha) Base (10 ha) Low (40 ha)

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

3

Cost ($/m ) 10.04 1.40 2.30 0.18

1.86 0.55 0.05

Gross merchantable volume (GMV) (m3/ha)

47.88 102.71 150.78 193.16 229.37 260.64 286.66 309.86 329.92 347.35 362.57

427.47 275.63 124.97

13.5858 9.5101 8.1515

Age-varying harvest costs for three cut block sizes Age class (years)

Woodlands cost ($/m3)

High case (5-ha) Base Case (10-ha) Low Case (40-ha) Harvest cost Harvest cost Harvest cost ($/m3) ($/m3) ($/m3)

10 20 30 40 50

13.95

29.73

28.42

27.92

60 70 80 90 100 110 120 130 140 150

12.88 11.89 11.16 10.64 10.26 9.97 9.75 9.58 9.44 9.33

28.66 27.67 26.94 26.42 26.04 25.75 25.53 25.36 25.22 25.11

27.35 26.36 25.63 25.11 24.73 24.44 24.22 24.05 23.91 23.80

26.85 25.86 25.13 24.61 24.23 23.94 23.72 23.55 23.41 23.30

Timber yields

Age class (years)

Establishment cost Water yield (106 $) (106 m3/103 ha)

Initial timber stocks Net saw wood volume (NSV) (m3/ha)

7.54 66.11 117.19 161.66 199.88 232.25 258.65 282.53 303.11 320.46 335.94

Age class (years) 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Initial stock (103 ha) 29.412 0.048 0.000 1.601 0.113 2.524 10.939 21.502 42.288 68.655 36.435 36.325 13.257 7.133 1.484

compel solutions in which the annual harvest meets or exceeds mill capacity and/or prohibits milling from varying widely. Many researchers utilize such constraints, e.g. Gunn and Rai [15], Gunn [13, 14], Moll and Chinneck [18], Reed and Errico [22, 23]. In contrast, Allard et al. [4] discuss how including timber demand functions and irreversible capacity investment in a forestry model eliminates the need for smoothing constraints. S1(T), S2(T), SP1(T) and SP2(T) specify our smoothing constraints and utilize parameter GAMMA. Measuring incremental water runo€ When a portion of a forest is harvested, the hydrology of the watershed containing the forest changes because harvesting alters the tree canopy and modi®es evapotranspiration (ET) characteristics. ET occurs when precipitation collects on the forest canopy and returns to the atmosphere directly without becoming water runo€. In snow-dominated forest areas, small cut blocks tend to retain more snow, reduce ET and generate more runo€ than do large cut blocks. However, a very large area typically must be harvested to signi®cantly a€ect stream¯ow, Swanson and Bernier [25]. In Table 3, water production constraints CAVOL(T) measure the volume of runo€ in period T as the water volume coecient V(J) multiplied by the total

282

John Rowse and Calum J. Center

hectares of all age classes harvested using method J.{ Persisting for only one period, AVOL(T) is incremental to the runo€ which occurs with no timber harvest. The V(J) are found using a water balance simulation model (see below). A brief note on timing is necessary. Adopting a common approach, we assume that harvesting and wood sales occur at the beginning of each period. However, per-period water produced V(J) actually occurs in 10 successive annual increments, v(J) = V(J)/10. Hence, coecient M is needed in PHI3 and PHI4 to transform the per-period runo€ value into the present value sum arising from selling the runo€ in 10 equal amounts at the water price VWAT.{ Concluding remarks on the model structure Implicitly, harvesting is assumed to be spatially dispersed. If a harvest strategy requires many small openings dispersed over a large area, then more roads must be built. Thus, the infrastructure or establishment cost of the road network must be speci®ed for each harvest method. Road construction costs are speci®ed by C(J), and road maintenance costs form part of MC1(J,A) and MC2(J,A). Infrastructure cost is assumed incurred at the outset. Finally, the saw wood price and pulp wood price are represented as PSAW and PPULP.

COMPUTATIONAL RESULTS Model data Our analysis area is a portion of the Bow-Crow forest on the eastern slopes of the Canadian Rocky Mountains of Alberta. Table 4 lists most of our model data. Timber yields GMV and NSV are for coniferous mixed wood, such as spruce, lodgepole, and Douglas ®r. Initial timber stocks S0(A) show that most timber is nearly a century old or older.} Not listed is TOTAREA of 271.736 103 ha. We assume that natural regeneration of coniferous trees occurs after harvest at the reforestation costs listed. That is, apart from replanting costs, no silvicultural costs are incurred following harvest. Product prices are constant for the entire 2.5 centuries, with saw wood at $200/m3 selling for about four times the $45/m3 price of pulp wood.} The water price of $0.08/m3 is based upon research by Thompson [26].|| Using research by Beck et al. [5], harvesting incurs costs that do not vary with tree age as well as those that do. Age-varying costs for each block size decline with age because costs per m3 are lower for larger trees. Costs are nil for timber younger than A50 because it is not merchantable.** {The CAVOL(T) constraints do not depend upon J because we solve our model for only one value of J at a time. {M is determined as follows. The left-hand side of the following equation is the product of M and per-period water value, yielding the right-hand side aggregated present value of ten sequential per-annum water ¯ows: M * 10 * v(J) * VWAT = v(J) * VWAT * [1 + 1/(1 + R) + 1/(1 + R)2+. . . + 1/(1 + R)9]. This simpli®es to: 10 * M = [1 + 1/ Thus M = [1 + 1/(1 + R) + 1/(1 + R)2 +. . . + 1/(1 + R)9]/ (1 + R) + 1/(1 + R)2+. . . + 1/(1 + R)9]. 10 9 10 = [(1 + R) ÿ1]/[10 R(1 + R) ]. }Forest inventory data is taken from the Alberta government's AFORISM database. Only good site index, mixed wood coniferous timber is used. Moreover, the GMV and NSV coecients are taken from Alberta Forestry, Lands and Wildlife [3]. }All selling prices are conjectural, of course, especially given the long time horizon employed. Steele [24] suggests that saw wood ranges in price from $80/m3 to $250/m3, and pulp wood ranges in price from $40/m3 to $50/m3. PSAW and PPULP fall within these ranges. ||Assigning a value to incremental water runo€ poses dicult problems because water markets in Alberta are not well developed. Thompson [26] ®nds that municipal, agricultural and industrial water use in Alberta are 8%, 75% and 17%, respectively. He determines water values per acre foot as follows: municipal, $173±$555; irrigation, $25±$154; and industrial, $98. Using a weighted average of the low and high values yields a range of $49.25 to $176.56 per acre foot, or $0.04 to $0.143 per m3. Our value lies within this range. Below we utilize $0.143/m3, the upper limit of this range. Two points should be noted. If incremental runo€ is used solely for irrigation, then VWAT should be $0.02/ m3 or lower. Further, we assume that the water is delivered to the user without cost. If transportation or distribution costs are incurred for water delivery, then VWAT should be lower. Consequently, our assumed water value might substantially overestimate the value of incremental runo€. **Age-varying costs of Table 4 are the sum of age-invariant costs and woodlands costs. For instance, for the 40-ha block size with a stumpage fee, the per-m3 age-varying cost for A50 timber is: $10.04 + $1.40 + $2.30 + $0.18 + $0.05 + $13.95 = $27.92. Other costs are found similarly.

Forest harvesting to optimize timber production and water runo€

283

What size cut blocks should be considered? Provincial guidelines allow blocks from 60 to 100 ha for harvesting pine and aspen, while spruce can be harvested in 24 ha patches or 32 ha strips. Alberta Environment Council [2] and Swanson and Bernier [25] conclude that commercial cut blocks in Alberta are typically well above the optimal size for snow catchment. Blocks in the 1to 10-ha range minimize snow scour (see Swanson and Bernier [25]). We consider three cut block methods and their associated road networks. Road costs are taken from Brown and Harding [10] and converted from 1982 US $/acre harvested to 1993 Cdn $/km. Bowes and Krutilla [7] utilize the same data for their road cost assumptions. The most extensive road network, for cut block method M1 (5-ha blocks), costs $427.47 million. The least expensive network, for method M3 (40-ha blocks), costs $124.97 million. Method M2 (10-ha blocks) assumes road costs of $275.63 million.{ Water coecients V(J) are found by running the water balance model WRNSHYD, which approximates ET losses, for di€erent cut block methods.{ First, the total area of good site index, mixed wood coniferous land is entered into WRNSHYD, then runs are made to calculate the incremental runo€ for each additional 104 ha harvested, up to one half of the analysis area, the maximum that can be cut in each period (Z = 0.5). For each block method the runo€ generated is observed to be a linear function of the area harvested; thus, the increment to water runo€ per 104 ha harvested is constant.} Cases considered We examine eight cases. For each block size, the model is solved without and with government stumpage fees, without and with water runo€ valued, and without and with harvest smoothing constraints. In particular, PHI1 through PHI4 are run ®rst without smoothing constraints, then with smoothing constraints. Succinctly, our cases are: Model

Characteristics

HARV1 HARV2 HARV3 HARV4 HARV5 HARV6 HARV7 HARV8

No stumpage fees, no water value, no harvest smoothing constraints Stumpage fees, no water value, no harvest smoothing constraints No stumpage fees, water valued, no harvest smoothing constraints Stumpage fees, water valued, no harvest smoothing constraints No stumpage fees, no water value, harvest smoothing constraints Stumpage fees, no water value, harvest smoothing constraints No stumpage fees, water valued, harvest smoothing constraints Stumpage fees, water valued, harvest smoothing constraints

For all cases, truncation is our terminal condition, a method that likely leads only to a small bias in model choice.} We also employ a discount rate of 5%. HARV1 and HARV2 For all block sizes, the HARV1 optimal harvest schedule is identical, a result also true for HARV2. Moreover, the HARV1 harvest schedule is identical to that for HARV2. {All block sizes assume a moderate slope and an average value between intermittent and seasonal road types. {WRNSHYD and its use are discussed in Center [11, App. C]. Model inputs consist of a water basin's hydrologic region, climate, vegetation, monthly precipitation, treatment description, total area under management, area harvested, windward width of the cuts, wind speed at ground surface, average height of surrounding trees, and basal area left after harvest. Units (or stands) can be speci®ed by hydrologic characteristic and many units can be simulated to estimate overall ET and runo€ for an area with many stands. Model outputs relate di€erent cut block size and shape arrangements to water yield and ET for an area. In assessing the accuracy of WRNSHYD, Bernier [6] ®nds that measured increases on experimental watersheds and WRNSHYD predictions of increases are very similar. }Incremental runo€ appears only for the period in which harvest occurs, but water production will likely vary within a period as the tree canopy reappears. We also assume that incremental water runo€ occurs for only one period, but additional runo€ may persist for more than a decade following harvest. }Terminal conditions may be important for determining the optimal solution. Truncation means that the future beyond the ®nal period of the model is ignored. This method is occasionally used in forestry models and is easily shown to involve very little bias to choice in our model when harvest smoothing constraints are absent and Faustmann-type results prove optimal (see Center [11, Ch. 6]). What happens when harvest smoothing constraints are present is less clear, since the harvest pattern becomes much more complicated. But the in¯uence of truncation upon net bene®ts in the presence of smoothing constraints is very small, as we show below.

284

John Rowse and Calum J. Center Table 5. Selected model outputs, HARV1 and HARV2 Timber yield (106 m3)

Harvest schedule (103 ha) Age class Time period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A70

Pulp wood

M1

M2

M3

36.064 23.278 0.013 0.188

3.770 3.032 0.004 0.054

1.846 1.422 0.002 0.022

1.292 0.996 0.001 0.015

1.107 0.853 0.001 0.013

0.048 29.412 135.858 104.684 0.113 1.601

0.006 3.447 15.921 12.268 0.013 0.188

0.002 0.988 4.563 3.516 0.004 0.054

0.001 0.400 1.846 1.422 0.002 0.022

0.000 0.280 1.292 0.996 0.001 0.015

0.000 0.240 1.107 0.853 0.001 0.013

0.048 29.412 135.858 104.684 0.113 1.601

0.006 3.447 15.921 12.268 0.013 0.188

0.002 0.988 4.563 3.516 0.004 0.054

0.001 0.400 1.846 1.422 0.002 0.022

0.000 0.280 1.292 0.996 0.001 0.015

0.000 0.240 1.107 0.853 0.001 0.013

0.048 29.412 135.858 104.684 0.113 1.601

0.006 3.447 15.921 12.268 0.013 0.188

0.002 0.988 4.563 3.516 0.004 0.054

0.001 0.400 1.846 1.422 0.002 0.022

0.000 0.280 1.292 0.996 0.001 0.015

0.000 0.240 1.107 0.853 0.001 0.013

2.524 0.113 1.601

>A70 135.858 102.160

Saw wood

Water runo€ (109 m3) Cut block size

Consequently, the $1.40/m3 stumpage fee has no e€ect on the optimal harvest pattern.{ Table 5 lists several time series common to HARV1 and HARV2.{ In the ®rst period, all A110 hectares and older are cut, as are 41.224 103 A100 hectares, for a total of 135.858 103 ha. In the second period, all A70 timber and older is cut, leaving all timber A60 and younger for harvest later. Table 5 also displays a recurring 70-year harvest pattern. Within the pattern, there is much variation from one period to the next. The HARV1 and HARV2 solutions warrant discussion. The model seeks a Faustmann solution, and the wood ®ber growth paths and product values e€ectively establish a 70-year rotation, making it optimal to harvest all trees aged 70 and older during the ®rst period. But the TOTHA(T) constraints (see Table 3) make such a harvest impossible. In the presence of the TOTHA constraintsÐbinding only for period 1Ðthe model harvests as much of the trees aged 70 and older during period 1 as possible, then defers the balance until period 2. Thereafter, the model harvests only A70 trees.} Timber and water volumes are listed in Table 5 and graphed in Fig. 1A and 1B. Immediately evident are the huge initial output of saw wood, not to be repeated, and the tremendous variation in outputs over time. The huge saw wood outputs stem from harvesting older trees, while the variation in timber and water production stems from the recurring harvest pattern. From the standpoint of economic stability, such widely varying timber volumes are undesirable and mill capacity might not be able to cope with such large variations. In addition, the varying

{This outcome is plausible for our LP model. However, including rising marginal cost curves for timber harvest and/or variable timber quality could cause the total harvest to be sensitive to the stumpage fee. {For HARV1 and HARV2, and the other solutions as well, we have made minor adjustments to eliminate obvious terminal e€ects. For instance, for all block sizes for HARV1 and HARV2, the model initially declared optimal a harvest of 0.048 103 ha of A50 timber in period 25, advancing by two decades what would have been the harvest of A70 timber in period 27 in a model with many more periods. To eliminate this e€ect, we re-ran the model with the A50 harvest set equal to 0 during period 25. The resulting changes to period 25 outcomes were minuscule andÐto six decimal digitsÐthere was no change to maximum net bene®ts. }To verify that the TOTHA constraints have this in¯uence, in a separate model run, Z was raised to 1.0, allowing all forested hectares to be cut in each period. All trees 70 years or older were harvested in period 1, and a 70-year rotation was found optimal. Replicating the Faustmann solution exactly, this result con®rms that our model is functioning correctly.

Forest harvesting to optimize timber production and water runo€

285

Fig. 1. (A) HARV1 saw wood and pulp wood production; (B) HARV1 water production.

runo€ volumes could be undesirable if they were large enough to a€ect stream¯ow.{ We note that the runo€ volumes are generated by the di€erent harvest methods, but for HARV1 and HARV2 these volumes play no role in determining the optimal solution. Table 6 lists HARV1 cumulative objective function values for the three cut block methods.{ M3 yields most bene®t, namely $9.123 billion, while M1 forgoes $0.406 billion, or 4.45% of maximum bene®t. M2 forgoes $0.179 billion or 1.97% of maximum bene®t. All net bene®ts exhibit a relatively small rise during the ®nal 15 periods, and thus harvests during the ®rst century account for nearly all net bene®ts. The huge saw wood and pulp wood yields during the ®rst two decades and discounting account for this outcome. Clearly, the relatively small rise {The mean annual water ¯ow for the South Saskatchewan River is 7.7 million acre feet [1] or 9.5 billion cubic meters. Harvesting using only the smallest block size and no smoothing constraints could yield an additional 0.065212 to 0.18 million cubic meters per year, with these amounts contributing an additional 0.0007% to 0.0019% of South Saskatchewan River ¯ow. Such percentage increments are obviously small. {Here and below we provide objective function values to six decimal digits for comparison purposes, not because we claim six-decimal-digit precision for our analysis. Some objective function values are identical to three decimal digits and hence more than three are needed for comparison purposes.

286

John Rowse and Calum J. Center Table 6. Cumulative objective function values (109$): HARV1

Period/method

M1

M2

M3

10 15 20 25

8.710262 8.714687 8.716153 8.716338

8.937138 8.941604 8.943084 8.943271

9.116437 9.120919 9.122405 9.122592

Table 7. Objective function values (109 $): HARV1±HARV4 Case/method

M1

M2

M3

HARV1 HARV2 HARV3 HARV4 HARV1* HARV3**

8.716338 8.636087 8.900498 8.820246 8.900498 9.046807

8.943271 8.863019 9.072183 8.991931 9.072183 9.173701

9.122592 9.042341 9.233088 9.152837 9.233088 9.320104

Note: *Water runo€ value included as a residual. **Water runo€ valued more highly at $0.143/m3

during the last ®ve periods means that any terminal e€ects are small, at least with respect to measured net bene®ts. HARV3 and HARV4 HARV3 includes the water production equations CAVOL(T) and represents water runo€ value of $0.08/m3 in the objective function. Including runo€ value does not change the optimal harvest pattern for all block methods from the pattern found optimal in HARV1 and HARV2. The optimal harvest patterns are identical for HARV4, implying that stumpage fees have no in¯uence either. Comparing HARV1 through HARV4 Table 7 lists objective function values for solutions HARV1 through HARV4 and two others. HARV2 is the same as HARV1, except that net bene®ts are reduced by stumpage. Stumpage of about $0.080 billion is the same for all blocks and stands at a little less than 1% of each objective function value. The stumpage fee is so small relative to other costs, PSAW and PPULP, that the optimal harvest pattern is insensitive to the fee (see Table 4). Although HARV1 does not include water runo€ value in net bene®ts, the value of incremental runo€ generated by timber harvest can be calculated and added to the HARV1 net bene®ts. The resulting values are listed as HARV1* in Table 7. These values are identical to those of HARV3 and still favour the largest block size, but the di€erence between M3 and M1 shrinks slightly to $0.333 billion. The value of runo€ as a residual accounts for about 2%, 1.4% and 1.2% of net bene®ts for the three cut blocks, respectively. Because the water value of $0.08/m3 does not in¯uence the optimal solution, lower values have no in¯uence either. But a higher water value might. From Bowes and Krutilla [7], we infer that the model might select a shorter rotation since a higher water value adds to the timber value and a higher timber price tends to reduce the rotation age. As a sensitivity analysis, we utilize a high water value of $0.143/m3.{ In this case, the model shifts to a 60- year harvest pattern for M1Ðthe method most favorable to water runo€Ðbut for M2 and M3 the 70-year harvest pattern remains optimal. The new objective function values are listed as HARV3**. The higher water value still does not make smaller cut blocks economic, but the di€erence in net bene®ts between M3 and M1 shrinks to $0.273 billion. HARV5 through HARV8 Characteristic of the HARV1 through HARV4 solutions are saw wood and pulp wood production levels that vary widely over time. If the harvester is an individual land owner or small producer, such variations are plausible. Yet, if the forest land harvested is large and/or the {See note k on pl 282 for the reason why this water value is chosen.

Forest harvesting to optimize timber production and water runo€

287

Table 8. Selected model outputs, HARV5 Harvest schedule (103 ha) Age class Time period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

A60

10.464 30.946 50.026 46.450 42.891 39.344 39.713 39.993 35.068 22.397 13.381 10.821 19.791 42.075

A70

13.425 35.237 34.673 33.630 32.965 30.397 16.373 2.226 1.198 0.466 0.001 2.082 7.325 17.184 25.252 29.976 28.524 19.922

>A70 47.480 43.935 40.010 33.899 30.397 26.837 19.747 15.987 12.243 9.262 6.380 0.934

Timber yield (106 m3) Saw wood 14.037 12.633 11.370 10.233 9.210 8.289 7.460 6.714 6.043 5.438 4.894 4.405 3.965 3.568 3.211 2.890 2.601 2.625 2.888 3.177 3.494 3.844 4.228 4.651 5.116

Pulp wood 1.248 1.123 1.011 0.910 0.819 0.737 0.811 0.892 0.981 1.079 1.187 1.306 1.436 1.580 1.738 1.585 1.440 1.453 1.534 1.530 1.397 1.338 1.403 1.543 1.698

Water runo€ (109 m3) Cut block size M1

M2

M3

0.645 0.597 0.544 0.461 0.413 0.365 0.451 0.696 0.637 0.583 0.535 0.568 0.643 0.710 0.647 0.589 0.535 0.540 0.572 0.576 0.538 0.525 0.554 0.656 0.842

0.452 0.418 0.380 0.322 0.289 0.255 0.315 0.487 0.446 0.408 0.374 0.397 0.450 0.497 0.453 0.412 0.374 0.378 0.400 0.403 0.376 0.367 0.388 0.459 0.590

0.387 0.358 0.326 0.276 0.248 0.219 0.270 0.418 0.382 0.350 0.321 0.341 0.386 0.426 0.388 0.353 0.321 0.324 0.343 0.346 0.323 0.315 0.333 0.394 0.505

analyst's perspective is that of a public land manager, wide variations may constitute doubtful public policy.{ Accordingly, we impose harvest smoothing constraints, as do other researchers; see Dykstra [12], Johnson [16] and Moll and Chinneck [18]. For HARV5 through HARV8, we solve HARV1 through HARV4 again, but in each period require saw wood and pulp wood production to be no greater than 110%, nor less than 90%, of the volume produced in the preceding period. Allowing only a 10% change between periods is arbitrary, but 10% has been used historically by other researchers. Further, we model two wood products, not one, and could utilize a di€erent percentage ®gure for saw wood than pulp wood if desired. Table 8 lists several outputs for HARV5. Old-growth timber is removed much more slowly than in HARV1: seven periods, rather than two, are required to remove trees A100 or older. Only by period 13, not by period 3, is the forest's oldest stand 70 years old. Following period 7, approximately 40,000±50,000 ha of land is harvested, and from periods 13 to 21, the majority of these ha yield A60 trees. The HARV5 harvest schedule is the same for all blocks.{ Table 8 lists the HARV5 volumes of saw wood, pulp wood and water runo€, and these are also graphed in Fig. 2A and 2B. The decline and rise of saw wood production are similar to the results of Reed and Errico [22] and Moll and Chinneck [18] and are easily explained. Large saw wood production occurs early because old-growth timber is cut as soon as possible, subject to the smoothing constraints. By the 8th period, the oldest stands are 80 years old, and by the 13th period the oldest stands are 70 years old. Saw wood volumes rise toward the horizon with larger A70-ha harvests. Pulp wood production and saw wood production adjust di€erently over time. The slight pulp wood production increase starting in period 7 and extending through period 15 occurs because progressively younger trees are harvested: a greater proportion of A60 and A70 trees are cut instead of trees A80 and older. If constraints SBAL(T) always bindÐthat is, the maximum {If public policy promotes continuing operation of saw mills and pulp mills for regional economic stability, and amenity values and wildlife habitat are important, then massive harvest variations over time most likely will not be tolerated by government. Moreover, because milling of saw wood or pulp wood is capital intensive, shutting down mills for prolonged periods of time would prove costly and inecient, and likely would not occur. Furthermore, wide swings in timber harvesting will lead to wide swings in the volumes of saw wood and pulp wood marketed. If the analysis area is large, then these wide swings could call into question our assumption that product prices are exogenous and constant throughout the model horizon. {One slight di€erence is that, in period 25 a truncation bias occurs: the model harvests all A60 timber rather than leaving some part of it for harvest as A70 timber in the next period. Because of discounting, however, the in¯uence of this terminal e€ect on net bene®ts is minuscule. We have made no adjustment to eliminate this terminal e€ect.

288

John Rowse and Calum J. Center

Fig. 2. (A) HARV5 saw wood, pulp wood and water production; (B) HARV5 hectares harvested and water production.

amount of saw wood is producedÐmore pulp wood is produced from harvesting younger trees than would be produced from harvesting older trees. For example, from Table 4, comparing GMV and NSV shows that for A150 trees, GMV = 362.57 m3/ha and NSV = 335.94 m3/ha, implying a pulp yield of 26.63 m3/ha. In contrast, for A70 trees, GMV = 150.78 m3/ha, NSV = 117.19 m3/ha, a di€erence of 33.59 m3/ha for pulp production. From period 12 onward the smoothing constraints also require harvesting A60 trees, which yield a GMV of 102.71 m3/ ha and an NSV of 66.11 m3/ha, leaving 36.66 m3/ha for pulp production. The decline in saw wood volume can also be explained by the `fall down' e€ect. If old growth timber predominates, then early harvests yield larger volumes per ha than subsequent harvests which occur much closer to the optimal rotation age. This fall down phenomenon occurs during the conversion to some normal forest state [20, p. 161].

Forest harvesting to optimize timber production and water runo€

289

Table 9. Cumulative objective function values (109 $): HARV5 Period/method

M1

M2

M3

10 15 20 25

5.131519 5.145578 5.146376 5.146473

5.328248 5.342446 5.343253 5.343350

5.496041 5.510292 5.511102 5.511199

With smoothing constraints imposed, an optimal rotation age no longer emerges because trees must be cut from several age classes simultaneously. The HARV5 solution moves toward a 70year rotation, but only by period 22 are A70 hectares the majority of hectares harvested. Water runo€ volumes still vary considerably over time. Figure 2B graphs water volumes for each of the three block methods and the total hectarage harvested (the latter being identical for each cut block method). Compared with Fig. 1B, Fig. 2B shows that water volumes need not track the volumes of timber harvested when a mix of age classes is harvested. Figure 2A illustrates a basic point: our LP model, embodying numerous linearity assumptions, generates di€erent and nonlinear time paths for saw wood, pulp wood and water production. In the presence of harvest smoothing constraints, wood volumes follow smooth but nonlinear paths, whereasÐbecause water volume depends upon hectares harvested, not timber outputÐwater runo€ ¯uctuates over time. These results are highly sensible and perhaps could not have been predicted a priori using economic intuition alone. Table 9 lists cumulative objective function values for HARV5. The large losses from the smoothing constraints are immediately evident. The most pro®table block size M3 garners bene®t of only $5.511 billion, 40% less than before. Of course, some losses could be illusory in that mill capacity might not sustain the HARV1 solution. As before, the values display a relatively small rise during the ®nal 15 periods. M3 is optimal and di€ers from M1 by $0.365 billion. If runo€ were valued, then its value would stand at 1.05% to 1.8% of the net bene®ts without water, similar to what was found before. Table 10 lists the objective function values for HARV5 through HARV8. Including a stumpage fee of $1.40/m3 generates harvest schedules for HARV6 identical to those for HARV5, while net bene®ts fall by about $0.048 billion (slightly less than 1.0%) for each block size. Valuing water runo€ at $0.08/m3 in HARV7 while excluding stumpage yields no changes to the harvest schedules but raises net bene®ts. In this case, the di€erence between M3 and M1 shrinks to $0.326 billion. Next, valuing water and including stumpage payments in HARV8 yield no changes in the harvest patterns from those of HARV5 but reduces all net bene®ts by about $0.048 billion. Further, if runo€ generated in the HARV5 solutions is valued as a residual and added to the HARV5 net bene®ts, then the HARV5* net bene®ts result. All are identical to those of HARV7, indicating that runo€ plays no role in determining the HARV7 solutions. Finally, if runo€ is valued more highly at $0.143/m3, then the HARV7** objective function values result. Here, the di€erence between M3 and M1 shrinks to $0.296 billion, but the increased water value still cannot justify the increased harvest costs of M1 relative to M3. Overall, comparing the objective function values of Tables 7 and 10 yields highly consistent outcomes. Without or with smoothing constraints, valuing water runo€ appears to hold little importance for optimal forest management. Table 10. Objective function values (109 $): HARV5±HARV8 Case/method

M1

M2

M3

HARV5 HARV6 HARV7 HARV8 HARV5* HARV7**

5.146473 5.098341 5.243242 5.195111 5.243242 5.319448

5.343350 5.295218 5.411088 5.362957 5.411088 5.464432

5.511199 5.463068 5.569261 5.521130 5.569261 5.614985

Note: *Water runo€ value included as a residual. **Water runo€ valued more highly at $0.143/m3

290

John Rowse and Calum J. Center

SENSITIVITY ANALYSIS How sensitive are our results? Assumptions about the values and costs of incremental runo€ are critical, but all are uncertain. Alberta water markets are regional and use-dependent: namely, residential, industrial and agricultural. Regional water demands may also be price-responsive and require specifying a demand function rather than just a price. Owing to the past abundance of water, Alberta markets for water are poorly developed and thus it is dicult to value incremental water supply with much certainty. Harvest costs are also uncertain, as are the di€erences across cut block methods. We assume that incremental water runo€ occurs during one ten-year period following timber harvest. Then, in the next period, tree growth causes runo€ to revert to what it was prior to harvest. This assumption is stark: the process of losing precipitation to ET actually unfolds continuously over time as trees grow, extends beyond ten years, depends upon the type of vegetation regenerated and can be slowed by vegetation management. Further, adopting harvest methods that leave slash as ground cover following harvest may enhance incremental runo€. We consider none of these alternatives. Many di€erent assumptions are plausible regarding the pulp wood and saw wood values and whether or not values change over time. Moreover, we conjecture that many di€erent assumptions will yield similar results. Quite di€erent assumptions could change the optimal rotation, whether runo€ is valued or not, and thus the maximum net bene®ts. But, we think that wood values will be secondary in importance to the assumed values and costs of incremental runo€. These latter elements are likely to prove so small relative to the net timber harvest bene®ts thatÐregardless of the wood pricesÐignoring incremental runo€ may prove optimal for forest management, or, at worst, ignoring runo€ is unlikely to be very costly in terms of bene®t foregone. For our sensitivity analyses we examine four scenarios. First, we alter our assumption that one-half of the analysis area can be harvested in each period. This assumption may be too generous relative to what forest managers would allow. We adopted it to avoid excessively constraining our solutions. To focus on more stringent cutting strategies, we allow only one-sixth of the area to be cut in each period (Z = 0.167), and then allow but one-®fth (Z = 0.20). The former assumption allows all of the analysis area to be cut in six decades, while the latter allows it to be cut in ®ve.{ For the remaining analyses we change the discount rate (R), ®rst to 2.5% and then to 6.0%.{ Table 11 summarizes our results here. To minimize solution details, we list only objective function values. For Z = 0.167, the recurring spikes in wood production of HARV1±HARV4 are much damped and net bene®ts shrink substantially from the base case (see Table 7). In contrast, the net bene®ts of HARV5±HARV8 are modi®ed only slightly (see Table 10). Overall, the consequences of valuing water runo€ are very similar. Raising Z to 0.20 yields slightly larger net bene®ts than for Z = 0.167, outcomes very similar and the interesting ®nding that the HARV5±HARV8 solutions are nearly identical to those of the base case.} Thus, the HARV5± HARV8 base case solutions are robust over virtually the entire range of Z = 0.2 through 0.5. Lowering R from 0.05 to 0.025 yields greater net bene®ts than the base case. However, larger relative increases are realized for the HARV5±HARV8 solutions than for the HARV1±HARV4 solutions, a highly sensible result. Moreover, the optimal rotation age rises from 70 to 80 years while the harvest patterns adjust accordingly. Raising R from 0.05 to 0.06 shrinks all net bene®ts from the base case while those of HARV1±HARV4 decline relatively less than those of {Although we vary Z for our sensitivity analyses, we do not change the V(J) coecients because our WRNSHYD runs suggest that the V(J) are constant for Z varying between 0 and 0.5. If the V(J) varied in a systematic way with Z, it would be easy to represent this relationship explicitly, but our model might be transformed into a nonlinear programming model as a result. {A discount rate upper limit of 6% was selected in part because Moll and Chinneck [18] make this choice, but also for practical reasons. Preliminary analysis with R = 0.075 led to numerical problems that served only to obscure the results of the sensitivity analysis. }Net bene®ts for HARV5±HARV8 di€er (in some instances) from the base case only in the sixth decimal digit, suggestingÐcorrectlyÐthat the corresponding solutions are very similar, but not identical.

Forest harvesting to optimize timber production and water runo€

291

Table 11. Objective function values for the sensitivity analyses (109 $) Z = 0.167 Case/method HARV1 HARV2 HARV3 HARV4 HARV5 HARV6 HARV7 HARV8

Z = 0.20

M1

M2

M3

5.321200 5.271588 5.421609 5.371997 5.046911 4.999677 5.141538 5.094302

5.519463 5.469851 5.589749 5.540137 5.242948 5.195714 5.309184 5.261949

5.687841 5.638229 5.748086 5.698475 5.410477 5.363243 5.467250 5.420015

Case/method HARV1 HARV2 HARV3 HARV4 HARV5 HARV6 HARV7 HARV8

M1

M2

M3

5.999102 5.943505 6.113793 6.058195 5.146472 5.098341 5.243242 5.195111

6.202965 6.147368 6.283249 6.227651 5.343349 5.295218 5.411088 5.362957

6.373482 6.317884 6.442296 6.38669 5.511199 5.463068 5.569261 5.521130

R = 0.025 Case/method HARV1 HARV2 HARV3 HARV4 HARV5 HARV6 HARV7 HARV8

R = 0.06

M1

M2

M3

10.421562 10.325800 10.673784 10.578023 8.006452 7.932808 8.182898 8.109254

10.663007 10.567245 10.839562 10.743801 8.227202 8.153558 8.350713 8.277069

10.847867 10.752106 10.999201 10.903439 8.404163 8.330519 8.510030 8.436386

Case/method HARV1 HARV2 HARV3 HARV4 HARV5 HARV6 HARV7 HARV8

M1

M2

M3

8.400682 8.323151 8.572479 8.494948 4.587095 4.543861 4.669879 4.626645

8.625068 8.547537 8.745285 8.667793 4.779390 4.736156 4.837338 4.794104

8.803418 8.725887 8.906492 8.828962 4.945491 4.902257 4.995161 4.951927

HARV5±HARV8. Simultaneously, the optimal rotation age shrinks to 60 years from 70. Valuing water runo€ leads to ®ndings identical to the base case. All sensitivity analyses yield the same qualitative outcomes as previously with regard to valuing water runo€. Our approach is deterministic, but water ¯ow variability may argue for a stochastic approach. Unfortunately, modelling stochastic elements involves numerous complexities. Many aspects of forest management involve uncertainty, such as ®re, drought, disease, seasonal and annual water ¯ow variability, timber and water values, and incremental runo€ as a function of cut block method. Further, modelling stochastic ¯ows likely requires considering water storage options and prospective ¯ood control and recreational bene®ts. Another issue is how to incorporate annual and seasonal ¯ow variability in a model with 10-year time periods. Although dicult, modelling stochastic water ¯ows is undertaken by Bowes et al. [8]. Given the small in¯uence we ®nd for valuing water runo€, we doubt that adopting a more complex stochastic approach would yield insights much di€erent than those presented here. We can imagine circumstances in which results might di€er, but we do not think that such circumstances are likely to occur. CONCLUDING REMARKS Motivating our work is this question: if the value of water runo€ enters into forest management decisions, can the incremental runo€ generated by modifying timber harvest practices meet or exceed the higher harvest costs? Our approach to this question builds upon previous forestry research using linear programming and utilizes a simulation model for watershed management. Some aspects of our work are uncommon or novel. First, we use two wood products whereas most forestry LP models consider only one. Second, we model water runo€ as a function of the hectares harvested, whereas most watershed models utilize prescription approaches ( [7] and [8]). Third, we solve for the optimal rotation age when water runo€ is valued rather than assessing individual timber stands for a number of di€erent possible rotations. We utilize an LP model with a 25-period 2.5 century time horizon, a saw wood price of $200/ m3 and a pulp wood price of $45/m3, plausible harvest and road costs speci®c to each of three cut block methods, and a 5% discount rate. When harvest smoothing constraints are absent and water runo€ is not valued, a 70-year rotation proves optimal. Valuing runo€ at $0.08/m3 has no e€ect on this rotation, while valuing runo€ more highly at $0.143/m3 shrinks the optimal rotation to 60 years, but only for the smallest cut block. Hence, optimal solutions that ignore water runo€ are almost completely insensitive to valuing runo€. Moreover, when smoothing constraints are present, valuing runo€ at $0.143/m3 still has no e€ect on the optimal harvest

292

John Rowse and Calum J. Center

pattern. Further, all harvest patterns are insensitive to stumpage. Finally, these ®ndings appear robust over a range of assumptions for di€erent parameters. These results con®rm the conjecture that valuing water runo€ is likely of little importance for optimal forest management. Similarly, valuing water runo€ as a residual from timber harvesting does not in¯uence optimal harvest decisions either. Our ®ndings might change if water values were much larger than foreseen, product prices were lower than foreseen, or strong societal preferences existed for smaller cut blocks because of amenities, wildlife habitat, improved forest access for ®ghting ®res or insect infestation or disease, or better emergency response-time access. Valuing these bene®ts monetarily might make smaller cut blocks socially preferred. Our approach ignores the issue of institutional and contractual arrangements facilitating harvest adjustment and the issue of water storage. However, our ®ndings support the ignoring of these issues because incremental runo€ values have very little impact upon optimal forest management. If we had found substantial gains from valuing both water and forestry products when choosing harvest patterns, then the dicult questions of realizing the gains through contractual agreements and developing water storage facilities would have to be addressed. In particular, if neighbouring political jurisdictions were involved, then devising the necessary institutional/contractual agreements would almost surely require public sector participation. We close with some remarks on forest harvest modelling. It may be desirable for other researchers examining the interaction of timber harvest methods and water runo€ to represent (at least) two wood products, rather than wood ®ber alone. The reason is that including water runo€ value may alter the optimal rotation age. Further, because of the nonlinear growth path of the gross merchantable volume of timber and the di€ering nonlinear growth path of saw wood, shifting to a di€erent rotation age would alter the relative outputs of pulp wood and saw wood. To allow these di€ering growth paths to in¯uence optimal forest management decisions properly, the amounts (and values) of saw wood and pulp wood may have to be represented explicitly. Representing wood ®ber alone would bias model results, the extent of which could be determined with further study. AcknowledgementsÐThis paper is based upon research reported more fully in Center [11]. We extend our sincere thanks to Richard Moll of Statistics Canada for his help with this work and to Richard, Dianne Draper and Elizabeth Wilman for helpful comments. We are responsible for any errors that remain.

REFERENCES 1. Alberta, Environment, Summary Report: South Saskatchewan River Basin Planning Program. Water Resources Management Services Planning Division, Government of Alberta, Edmonton, 1984. 2. Alberta Environment Council, Our dynamic forests: The challenge of management. Environment Council of Alberta, Edmonton, 1990. 3. Alberta, Forestry, Lands, and Wildlife, Alberta phase 3 forest inventory: yield tables for unmanaged stands. Alberta Energy and Natural Resources, Government of Alberta, Edmonton, 1985. 4. Allard, J., Errico, D. and Reed, W., Natural Resource Modeling, 1988, 2, 581±597. 5. Beck, J., Anderson, R., Armstrong, G. and Farrow, G., Alberta economic timber supply analysis: Final report. Department of Forest Science, University of Alberta, Unpublished manuscript, 1988. 6. Bernier, P., in Forest modeling symposium, ed. B. Boughton and J. Samoil. Information Report NOR-X-308. Northern Forestry Centre, Forestry Canada, pp. 120±127, 1990. 7. Bowes, M. and Krutilla, J., Multiple use management: The economics of public forest lands. Resources for the Future, Washington, DC, 1989. 8. Bowes, M., Krutilla, J. and Stockton, T., in Emerging issues in forest policy, ed. P. Nemetz. University of British Columbia Press, Vancouver, pp. 370±406, 1992. 9. Brooke, A., Kendrick, D. and Meeraus, A., GAMS: a user's guide. Scienti®c Press, Redwood City, CA, 1988. 10. Brown, T. and Harding, B., in Management of subalpine forests: Building on 50 years of researchÐProceedings of a technical conference, ed. C. Troendle, M. Kaufman, R. Hamre and R. Winokur. USDA Forest Service, Colorado, 1987. 11. Center, C. Optimizing timber and water production using linear programming. Unpublished M.A. thesis. Department of Economics, University of Calgary, 1994. 12. Dykstra, D., Mathematical programming for natural resource management, McGraw-Hill, New York, 1984. 13. Gunn, E., Some aspects of hierarchical production planning in forest management. Working Paper ]91-04, Department of Industrial Engineering, Technical University of Nova Scotia, 1991. 14. Gunn, E., Hierarchical planning processes in forestry: Some perspectives from stochastic programming. Working Paper ]92-05, Department of Industrial Engineering, Technical University of Nova Scotia, 1992. 15. Gunn, E. and Rai, A., Canadian Journal of Forestry Resesarch, 1987, 17, 1507±1518.

Forest harvesting to optimize timber production and water runo€

293

16. Johnson, K., FORPLAN version 1: an overview. Land Management Planning Systems, USDA Forest Service, Fort Collins, 1986. 17. Moll, R., Modeling regeneration and pest control alternatives for a forest system in the presence of ®re risk. Unpublished Ph.D. dissertation, Department of Systems and Computer Engineering, Carleton University, 1991. 18. Moll, R. and Chinneck, J., Natural Resource Modeling, 1992, 6, 23±49. 19. Navon, D., Timber RAM: a long-range planning method for commercial timberlands under multiple-use management. Paci®c Northwest Forest And Range Experiment Station, USDA Forest Service, Berkeley, 1971. 20. Pearse, P., Introduction to forestry economics. University of British Columbia Press, Vancouver, 1990. 21. Reed, W., Natural Resource Modeling, 1986, 1, 55±75. 22. Reed, W. and Errico, D., Canadian Journal of Forestry Research, 1986, 16, 266±278. 23. Reed, W. and Errico, D., Natural Resource Modeling, 1989, 3, 399±427. 24. Steele, E., Personal Communication to C. Center (Nov. 24, 1993). Forest Alliance of British Columbia, Vancouver, 1993. 25. Swanson, R. and Bernier, P., in Proceedings of the Canadian hydrology symposium No. 16: DroughtÐThe impending crisis?. National Research Council: Associate Committee on Hydrology, Ottawa, pp. 485±496, 1986. 26. Thompson, J., in Flowing to the future: Proceedings of the Alberta Rivers Conference, May 11±13, 1989, ed. C. Bradley, A. Einsiedel, T. Pyrch and K. Van Tighem. Faculty of Extension, University of Alberta, 1990. 27. Weintraub, A. and Navon, D., in TIMS studies in the management sciences 21, ed. M. Kallio et al. North-Holland, New York, pp. 337±351, 1986.