Forestry An International Journal of Forest Research Forestry, Vol. 85, No. 4, 2012. doi:10.1093/forestry/cps043 Advance Access publication date: 20 May 2012 Anticipatory vs adaptive optimization of stand management when tree growth and timber prices are stochastic TIMO PUKKALA* and SEPPO KELLOMÄKI University of Eastern Finland, School of Forest Sciences, PO Box 111, 80101 Joensuu, Finland * Corresponding author. E-mail: timo.pukkala@uef.fi Summary Forest management involves considerable amounts of uncertainty related to future timber prices and tree growth. A new element in forest-management planning is climate-induced change in tree growth. This study used stochastic adaptive optimization to derive optimal adaptive rules for the management of a mixed stand of Norway spruce, Scots pine and birch, when (1) both price and growth were stochastic and (2) there was a climate-induced growth trend. Optimized reservation-price function was used as the adaptive rule for final felling. The optimal times of thinning treatments were described by rules that related the thinning year to the growth rate of the stand. The results suggest that an improving growth trend slightly shortens optimal rotation lengths. On the other hand, risk related to timber price and growth tended to increase the rotation length. Increasing timber-price volatility increased the reservation price and rotation length. When risk and risk-aversion increased, maintaining a more diverse stand structure was profitable. Introduction Management instructions for tree stands are often expressed as recommended rotation lengths, thinning years and thinning intensities. In many cases, recommendations are supported by analyses in which the cutting years are optimized to maximize timber production or net present value (NPV; Valsta, 1992a, 1992b; Pukkala et al., 1998; Vettenranta and Miina, 1999; Hyytiäinen and Tahvonen, 2002; Hyytiäinen et al., 2004). However, if true growth differs from the assumed growth rate, for instance due to climate change, management based on this type of instructions may be non-optimal. Different rotation lengths and cutting years are often recommended for different growing sites. Optimal rotations tend to be shorter when growth is better (Palahí and Pukkala, 2003). However, the situation is more complicated when there is a growth trend. A steadily improving growth may maintain a certain relative value increment for a longer time and the optimal relative value increment rotation may even increase as compared with a constant, non-improving growth rate. One possibility to specify a management schedule under changing growth conditions is to use decision variables that depend on the achieved growth: thinning years may be replaced by thinning basal areas, and rotation length replaced by the tree diameter required for clear cutting. This has happened, for instance, in Finnish management instructions (Anonymous, 2006); but the main reason for the change is that basal area and mean diameter are easier to measure precisely in the forest than in the stand age. The problem of specifying a management schedule by thinning basal areas and clearfelling diameters is that the optimal thinning basal area or clearfelling diameter may also depend on the actual growth rate of trees; it seems logical to postpone thinning if trees are still growing well. The mean tree diameter at clearfelling is most probably greater when growth is good. Allowing the stand to continue growing for an additional year is more profitable with an improving growth trend than without it. An obvious solution to this dilemma is to replace the fixed cutting years, basal areas or cutting diameters by functions in which these parameters depend on the actual Institute of Chartered Foresters, 2012. All rights reserved. For Permissions, please email: journals.permissions@oup.com.
464 FORESTRY growth and stand development. The task of optimization is now to find the best possible parameter values for these functions; rules for adaptive cutting are optimized instead of following exact cutting years (Lohmander, 2007). These functions help forest landowners to adapt forest management on the basis of the realized true growth instead of following instructions that are based on certain anticipated growth. In this article, optimization that produces fixed cutting years, cutting basal areas or cutting diameters is called anticipatory optimization. Optimization that produces rules on how to react to the actual state of nature is called adaptive optimization (Lohmander, 2007). Both optimization approaches lead to the same management strategy when growth and all other factors are deterministic. Differences appear when growth is stochastic. Timber prices also vary, which means that the optimal cutting year is not always the same, even if growth is deterministic. The problem of stochastic timber prices is conveniently solved by using a reservation-price function (Brazee and Mendelsohn, 1988; Lohmander, 1995; Gong and Yin, 2004). This function shows how the selling price depends on stand age, volume or mean tree diameter; if the actual timber price is lower than the reservation price, a forest landowner should not cut but wait until the actual price increases or the reservation price decreases. Reservation price usually decreases with increasing stand age or tree diameter (Lu and Gong, 2003). The task of adaptive optimization under stochastic timber prices is to find the optimal values of the parameters for the reservation-price function. Previous studies (Pukkala, 2006) indicate that the timing of final felling is clearly more sensitive to timber price than the timing of thinning. This means that it is especially important to develop reservation-price functions for final felling. Under uncertain growth and economic conditions, optimization of forest management should produce rules that would allow forest landowners to adapt the management to changing situations. The aim of this study was to show how this can be done when both growth and price are stochastic. Another objective was to estimate the gains obtained when anticipatory optimum is replaced by adaptive optimum. The management implications of stochastic growth and timber prices are also examined. The article also discusses the influence of a forest landowner s risk attitude on optimal management. Methods Transfer of climate change into growth model The response to climate change was assumed to be species-specific, specific to maturity (size) of trees, tree spacing, position of trees in the stand and site fertility; see Kellomäki et al. (2012) for details. Consequently, Growth = f (climate change, tree species, maturity ( H, D), spacing, position, site type) where D is the diameter and H the height of a tree. The impacts of climate change were described as relative changes in growth (ΔG(REL)) caused by the change in climate (temperature, precipitation and CO 2 ), as follows: D(CC) G(REL) = D(CUR) where CC refers to growth under climate change and CUR to growth under the current climate in otherwise similar conditions. The ratio between the growths under climate change and current climate provides a correction factor, which is used to correct the predicted radial and vertical growth when climate change is assumed. The process-based model FinnFor (Kellomäki and Väisänen, 1997; Ge et al., 2010) was used to generate data for the correction factors. The model utilizes the main physiological processes (photosynthesis, respiration, transpiration and water and nutrient uptake) for calculating the growth of trees. The model has been parameterized for the main tree species in Finland (Pinus sylvestris, Picea abies, Betula pendula and Betula pubescens). The time resolution of the model is 1 h. In the calculations, the initial tree stands represented different combinations of tree species of varying sizes growing in stands of varying spacing and site fertility. The trees of a stand were divided into three size cohorts representing different tree positions. The time series of climate scenarios (current and changing climates) were cut in short slices (5 years), which were used to calculate the response of each initial tree stand to the current and changed climates during each time slice. This made it possible to identify how young and mature trees in different positions and spacing respond to changes in climate on poor and fertile sites. The calculations were carried out for Scots pine, Norway spruce and birch under the conditions of the northern boreal case study area used in the Models for Adaptive Forest Management (MOTIVE) project (http:// motive-project.net/). The current climate was that for the Joensuu airport (62 40 N, 29 38 E, a.s.l.: 94 m) during 1960 2000. The atmospheric mean CO 2 concentration was 363 ppm for the current climate, and it increased to 635 ppm by the end of the simulation period following the climate-change scenario based on the emission scenario A1B (a balance across all sources) of the Special Report on Emissions Scenarios (SRES) (Intergovernmental Panel on Climate Change or IPCC, 2007). Table 1 shows that the mean annual temperature increased from 2 to 6 C and the mean temperature in summer (June, July and August) increased from 14 to 17 C. At the same time, the annual mean precipitation increased from 500 to 550 mm; and the mean summer precipitation (June, July and August) increased by 7 per cent. The climate change increased the radial and vertical growth of Scots pine from 0.36 per cent per year up to 0.59 per cent, in such a way that the relative increase was larger on poor sites compared with that on fertile sites (Table 2). The same pattern held for Norway spruce and birch, but (1)
Table 1: Current and changing climate during selected time periods Current and periodical mean values under climate change OPTIMIZATION OF STAND MANAGEMENT 465 Mean temperature (standard deviation, C) Mean precipitation (standard deviation, mm) Annual Summer Annual Summer Mean CO 2 concentration (ppm) Current 2.3 (0.6) 14.1 (1.0) 493 (37) 199 (44) 363 2010 2040 3.1 (0.7) 14.7 (1.0) 506 (34) 203 (42) 412 2040 2070 4.6 (0.7) 15.7 (1.0) 529 (35) 208 (43) 507 2070 2100 6.2 (0.8) 16.8 (1.0) 554 (36) 213 (44) 635 Summer refers to June, July and August. Table 2: Relative changes in radial and vertical growth as a function of tree species and site type (Kellomäki et al., 2011) Species Site type Change in growth per year (%) Scots pine OMT 0.362 MT 0.414 VT 0.523 CT 0.596 Norway spruce OMT 0.266 MT 0.293 Birch OMT 0.328 MT 0.334 Site type refers to the site fertility: OMT = herb rich; MT = mesic; VT = sub-xeric; CT = xeric. in these cases, the increase was smaller than that for Scots pine. The increase was larger for birch than for Norway spruce. Optimizations Management was optimized for a planted spruce stand growing on a medium site (MT in Table 2). Regeneration was predicted using the models of Miina and Saksa (2006). Their models predict the numbers of planted spruce, natural spruce, natural pine, natural birch and hardwood coppice at the time when the planted spruces are 3 years old. In all optimizations, a tending operation was simulated at 10 years, removing all hardwood coppice and leaving a mixed stand of spruce, pine and birch (750 spruces, 570 pines and 750 birches per hectare). The aim was to leave the same number of seedlings of each species, but there were not enough pines for this. This type of mixed, young-sapling stand would allow many future management options in terms of species composition, depending on the growth and price of different tree species. In growth simulations, the planted species (spruce) was described with three cohorts: small, medium and large spruce seedlings, representing 170, 250 and 330 trees per hectare, respectively, after the tending operation at 10 years. Pine and birch were represented by one cohort each. Harvest percentages were optimized separately for different cohorts. A two-thinning schedule was optimized in most analyses. The growth and survival of trees were simulated using the models of Hynynen et al. (2002); but the radial growth predictions were multiplied by the climatechange-related correction factor. Autocorrelated stochastic annual variation was also added to the predictions using the method explained below. Assortment volumes of removed trees were calculated with the taper models of Laasasenaho (1982). The roadside prices of timber represented the average prices during the previous 5 years and were 60 m 3 for pine and spruce log, 50 m 3 for birch log and 30 m 3 for pine, spruce and birch pulpwood. The harvesting costs were calculated using the models of Valsta (1992a). The stand establishment cost at year 0 was 1042 ha 1 and the tending cost at year 10 was 564 ha 1, based on the number and size of the removed trees. Three alternative problem formulations were tested. They differed in the decision variables that were used to specify the times of thinnings and final felling. The decision variables of the three formulations were as follows: 1 Formulation 1 (anticipatory) a thinning years b harvest percentages c clearfelling year 2 Formulation 2 (anticipatory, timing depends on growth) a thinning basal areas b harvest percentages c clearfelling diameter 3 Formulation 3 (adaptive) a growth-adjusted thinning years b harvest percentages c reservation-price function for clearfelling The Nelder Mead method was used in optimization (Nelder and Mead, 1965; Pukkala, 2009). NPV of all costs and incomes was maximized with a three per cent discount rate. Optimization with the scenario technique (Valsta, 1992b) was used when growth and timber price were assumed stochastic. Five cases in terms of growth, price variation and risk preferences of the forest landowner were analysed: 1 Effect of growth trend: deterministic growth and price, without and with trend and risk-neutral owner (using all three formulations)
466 FORESTRY 2 Effect of growth variation: stochastic growth, deterministic price, growth trend and risk-neutral owner (all formulations) 3 Effect of timber-price variation: deterministic growth, stochastic price, growth trend and risk-neutral owner (formulations 1 and 3) 4 Effect of growth and price variation: stochastic growth and price, growth trend and risk-neutral owner (formulations 1 and 3) 5 Effect of risk attitude: stochastic growth and price, growth trend, risk-averse, risk-neutral or risk-taking owner (formulation 3). Stochastic annual variation in tree growth was produced by using the models of Pasanen (1998). Stochastic timber-price scenarios were produced using the models of Leskinen and Kangas (1998). Both models take into account the positive autocorrelation between successive years and the positive cross-correlation between the growth patterns of the tree species or between the prices of different timber assortments (Figures 1 and 2). Accordingly, 100 growth and price scenarios were generated in every optimization in which growth or price was stochastic (Cases 2 5). Growth variation was generated around the growth trend and the trend was also assumed stochastic; the slope of the trend was multiplied with a random number having mean equal to 1 and standard deviation equal to 0.2. Every schedule that was evaluated during the optimization process was simulated 100 times. The expected NPV was calculated as the mean NPV of 100 simulations. In Case 5, where risk attitude was examined, the 10th, 50th and 90th percentiles of the distribution of NPV were calculated. In problem formulation 3 (adaptive optimization), the timing of final felling was determined by a reservationprice function, which was optimized. The stand was clearfelled once the volume-weighted mean roadside price of timber assortments exceeded the reservation price. The reservation-price function had a shape similar to that found suitable in earlier studies (Lohmander, 1995; Brazee and Bulte, 2000; Lu and Gong, 2003; Pukkala, 2006), which is represented as follows: (a) (b) Figure 1. Example growth scenario for pine, spruce and birch (a) and five growth scenarios with trend lines for spruce (b). P = exp( A 0.6ln( D )) (2) where P is the reservation price ( m 3 ), D the basalarea-weighted mean diameter (cm) and A a variable to be optimized. The constant, 0.6, was based on preliminary optimizations in which both constant A and the coefficient for ln(d) were optimized; 0.6 was found to be a suitable overall slope because the optimal values were near 0.6 in all cases. Using a fixed slope coefficient made the optimization tasks easier. In adaptive optimization (formulation 3), a fixed thinning year or thinning basal area was replaced by growthadjusted thinning year. The year of the first thinning was Y/M B, where M is the multiplier of the slope of the growth trend (N(1,0.2)) and B is a parameter to be optimized together with Y. Here, Y is the thinning year under average Figure 2. A timber-price scenario. growth trend. If there was another thinning, the number of years up to the second thinning was I/M B, where I is the number of years between the first and the second thinnings, which was one of the optimized decision variables. If the value of B is larger than zero, better growth results in earlier thinning.
OPTIMIZATION OF STAND MANAGEMENT 467 Table 3: NPV, optimal rotation length, thinning basal area and clearfelling diameter for different problem formulations when growth and timber price are deterministic Decision variables NPV ( ha 1 ) Rotation (years) Basal area at first/second thinning (m 2 ha 1 ) Mean diameter at clearfelling (cm) No growth trend Years* 1474 66.4 22.3/35.6 27.8 Stand status 1445 66.5 23.4/36.9 28.3 Reservation price 1452 65.9 19.7/34.5 28.1 Growth trend (improving growth) Years* 2108 60.4 18.3/36.0 28.2 Stand status 2058 64.5 24.9/39.1 28.6 Reservation price 2064 60.9 20.7/39.6 27.5 * Thinning and clearfelling year optimized with harvest percentages. Thinning basal area and clearfelling diameter optimized with harvest percentages. Growth-adjusted thinning years and reservation-price function optimized with harvest percentages. Basal-area-weighted mean diameter. Table 4: Expected NPV, rotation length, thinning basal area and clearfelling diameter for different problem formulations with stochastic growth, climate-induced growth trend and deterministic timber price Decision variables NPV ( ha 1 ) Rotation (years) Basal area at first/second thinning (m 2 ha 1 ) Mean diameter at clearfelling (cm) Years* 2071 63.8 21.3 25.0/33.4 40.5 26.8 29.1 Stand status 2026 59.3 70.4 26.2/40.5 27.4 Reservation price 2076 58.8 68.6 17.8 23.4/41.2 47.1 27.4 29.6 * Thinning and clearfelling year optimized with harvest percentages. Thinning basal area and clearfelling diameter optimized with harvest percentages. Growth-adjusted thinning years and reservation-price function optimized with harvest percentages. Basal-area-weighted mean diameter. Results Effect of growth trend When both growth and timber price are deterministic, all the three problem formulations should result in similar optimal management. Figure 3 shows that this was roughly the case, although there were some slight differences in optimal rotation lengths, thinning basal areas and clearfelling diameters (Table 3). The optimal management involves removing practically all the birch in the first thinning and mainly pines in the second thinning. The slight differences between problem formulations are due to technical reasons such as the inability of the optimization method to find the exact global optimum and the fact that the time step of the used growth models being 5 years, interpolation was needed when the optimal thinning basal area or clearfelling diameter was reached during a 5-year period. An improving growth trend seems to slightly shorten the optimal rotation length and increase the basal area at which the stand is thinned for the second time (Table 3). However, differences were minimal, implying that the optimal schedule obtained without considering a growth trend would also be good under improving growth. The growth trend would increase the NPV by 42 per cent and the mean annual harvest would increase from 7.60 to 8.85 m 3 ha 1 (by 14 per cent). Effect of stochastic growth All the three problem formulations were also tested under stochastic tree growth. Annual growth variation and variation in the slope of the growth trend were assumed. Problem formulation 1 produces constant cutting years but variable thinning-basal areas and clearfelling diameters. Formulation 2 produces constant cutting-basal areas and clearfelling diameters but variable thinning years and rotation ages. In formulation 3, all management parameters (rotation length, thinning basal area and clearfelling diameter) depend on the actual growth (Table 4). There was no large difference in the expected NPV (mean NPV of 100 growth scenarios) between the two anticipatory and the one adaptive problem formulations. The optimal solution obtained with the anticipatory approaches (formulations 1 and 2) would be equally good as the adaptive optimum. The results indicate that when tree growth is assumed stochastic, optimal rotation lengths are slightly longer, thinning basal areas slightly higher and clearfelling diameters slightly larger, as compared with
468 FORESTRY Figure 3. Optimal management schedule without (top) and with (bottom) growth trend and with three different problem formulations. Formulation 1 = cutting years optimized; formulation 2 = thinning basal areas and clearfelling diameter optimized; formulation 3 = thinning years and reservation-price function optimized. Figure 4. Expected NPV obtained in anticipatory and adaptive optimizations with different amounts of stochastic variation in timber price and tree growth. optimal management with deterministic growth (Tables 3 and 4). Under stochastic growth, some more birch is left to continue growing in the first thinning and more pines are left in the second thinning. Therefore, the optimal species distribution of the stand at final felling will be uniform; a mixture of all species should be maintained when there is uncertainty about future tree growth. However, all the Figure 5. Optimal reservation-price functions with different amounts of annual variation in timber price. differences in management prescriptions obtained from deterministic and stochastic optimizations were small. Effect of stochastic timber prices In Case 3, cutting years were optimized in the anticipatory approach (formulation 1) whereas growth-adjusted thinning years and reservation-price function for final felling
OPTIMIZATION OF STAND MANAGEMENT 469 was 96 per cent (81 per cent in the second thinning) with a constant timber price, 84 per cent (68 per cent) with normal price variation and 60 per cent (49 per cent) with doubled price variation. As a consequence, the share of birch at final felling increased from 1 to 22 per cent when price variation increased from zero to double-normal. Figure 6. Expected NPV obtained for one- and two-thinning schedules in anticipatory and adaptive optimizations when both timber price and tree growth are stochastic. Figure 7. Dependence of optimal thinning years on the achieved growth level in a two-thinning schedule. Multiplier 1 (x-axis) means that growth follows the anticipated climate-induced growth trend. Smaller multipliers indicate slower growth (flatter trend) and higher multipliers, better growth (steeper trend). were optimized in the adaptive approach (formulation 3). When timber price was assumed stochastic and growth was deterministic, the benefit obtained from adaptive optimization and management was clearly higher than that under stochastic growth (Figure 4). Increasing timber-price variation increased the superiority of the adaptive approach. The reservation price increased with increasing annual variation in timber price (Figure 5); with high price variation, the minimum selling price was higher and it was worthwhile to wait for a good price for a longer time than with small price variation. The actual rotation lengths obtained from adaptive optimization vary more when price fluctuations are high. The mean rotation length of the optimal management schedule was 63.3, 69.7 and 76.0 years, respectively, when timber price was constant, varied similarly as in the past, or twice as much as in the past. The corresponding standard deviations of the rotation lengths were 0.0, 5.9 and 11.6 years. Increasing uncertainty related to future timber prices shifted the optimal stand structure towards a mixed stand. The optimal thinning intensity of birch in the first thinning Effects of stochastic growth and price When both growth and timber price were stochastic and represented the normal levels of annual variation, adaptive optimization (formulation 3, growth-adjusted thinning years and reservation-price function optimized) resulted in clearly higher expected NPV than the anticipatory approach (formulation 1, cutting years optimized). Improvement in the expected NPV was 24 per cent (Figure 6). When a onethinning schedule was optimized, the adaptive approach resulted in 13 per cent higher expected NPV. The optimal two-thinning management schedule for adaptive management can be described as follows. 1 Thin the stand at the age of 30.2/M 0.268 years, where M is the ratio between actual growth trend and average trend (better growth leads to earlier thinning). Remove 45 per cent of small spruces, 6 per cent of mediumsized spruces, 2 per cent of large spruces, 28 per cent of pines and 84 per cent of birch. 2 Thin again after 22.6/M 0.268 years. Remove 63 per cent of small spruces, 29 per cent of medium-sized spruces, 17 per cent of large spruces, 30 per cent of pines and 68 per cent of birch. 3 Clearfell once the mean roadside prices of all trees of the stand are higher than the reservation price (P) calculated from P = exp(5.956 0.6ln(D)) where D is the basal-area-weighted mean diameter (cm). Figure 7 shows that the thinning years are earlier when growth is good. The first thinning should be conducted later than suggested by the anticipatory optimum. The reservation price for final felling is higher than that in the case when only timber price is stochastic (Figure 5, thick solid line and thin dashed line). Effect of risk attitude The effects of landowner s risk preferences were studied using the adaptive approach (formulation 3) under normal variations in growth and price. The risk-avoider maximized the 10 per cent accumulation point of the distribution of NPV, the risk-neutral owner maximized the 50 per cent accumulation point and the risk-seeker maximized the 90 per cent accumulation point. The risk-avoider wanted to find a management that is the best under unfavourable growth conditions and price development, whereas the risk-taker sought management that is the best under favourable growth conditions and price development. The differences between the risk attitudes in forest management were several but rather small (Table 5). The risk-avoider thins first and the risk-taker, last. In the optimal
470 FORESTRY Table 5: Optimal adaptive two-thinning schedules for planted spruce under stochastic growth and price and improving growth trend Management parameter management for a risk-avoider, the discounted net incomes are more evenly shared among the three cutting events than in the optimal schedules for the other risk attitudes. The risk-avoider maintains the most balanced species composition (Figure 8); the risk-neutral forest landowner leaves less pine than the risk-avoider and the risk-taker removes practically all birches in the first thinning. The reservation-price function is nearly the same for all risk attitudes. The distribution of different timber assortments at clearfelling was more uniform for the risk-avoider than for the risk-neutral and the risk-taker landowners (Figure 9). The risk-taker removed, almost exclusively, birch pulpwood in the first thinning and mainly spruce in the final felling. The average mean annual removal was 8.2 m 3 ha 1 for the riskavoider, 8.5 m 3 ha 1 for the risk-neutral landowner and 8.9 m 3 ha 1 for the risk-seeker. The risk-sharing behaviour of both risk-avoider and risk-neutral landowners decreased the yield compared with the risk-seeker who concentrates the production to conifers and the incomes to the final felling. Discussion Risk-avoider Risk-neutral owner Risk-taker Stand age at first 30.9/M 0.208 34.7/M 0.221 35.2/M 0.228 thinning Harvest percentages Small spruce 47 42 43 Medium spruce 18 5 4 Large spruce 8 3 9 Pine 39 50 34 Birch 66 65 91 Years to second 19.6/M 0.208 16.1/M 0.221 20.0/M 0.228 thinning Harvest percentages Small spruce 48 48 56 Medium spruce 28 17 21 Large spruce 26 15 25 Pine 34 54 35 Birch 63 55 55 Reservation price exp(6.056 0.6ln(D)) exp(6.077 0.6ln(D)) exp(5.999 0.6ln(D)) M = multiplier of the growth trend ( growth level ); D = the basal-area-weighted mean diameter (cm). The stand is clearfelled when the actual mean roadside price ( m 3 ) of all trees exceeds the reservation price. The method used in this study corresponds to optimization of adaptive control functions with stochastic simulation of the objective function in the classification of Lohmander (2007). Its advantage is flexibility, but the number of calculations is high. Strictly speaking, our analysis was a combination of adaptive and anticipatory optimization approaches because adaptive rules were developed only for the cutting years whereas thinning intensities were based on the anticipatory approach. The study is not the first one to examine optimal control functions for adaptive forest management. However, most of the previous studies have concentrated on price variation. Very few analyses have incorporated price and growth volatility (Penttinen, 2006), mixed stands and growth trend. In addition, one of our analyses also considered the risk preferences of forest landowners, which have been found to have a significant impact on optimal forest management (Pukkala, 1998). In the stand analysed in this study, deterministic optimization that omitted the growth trend gave almost the same optimal management as the deterministic optimization in which the improving growth trend was considered. The main difference was that the cuttings tended to be a few years earlier when a gradually improving growth was assumed. This is in line with earlier results, which show that improving growth shortens optimal rotations (Palahí and Pukkala, 2003). However, our results cannot be generalized to all species and stands because constantly improving growth can maintain a certain relative value increment for more years than non-improving growth, which counteracts the overall influence of growth rate on the optimal timing of cuttings. Using adaptive optimization, instead of the anticipatory approach, did not offer big benefits when only the growth was stochastic. Similar to earlier research (Gong, 1998; Lu and Gong, 2003), when timber price was assumed to vary, adaptive optimization (followed by adaptive management) led to clearly better profitability than anticipatory optimization (followed by non-adaptive management). The advantage of the adaptive approach would be higher if the simple cutting rules used in this study were replaced by ones that are more complicated. For example, it would be possible to use reservation price also in thinning, and the reservation-price function could include more than one stand variable (e.g. both mean diameter and stand basal area); moreover, the actual growth rate of trees could also be used as a predictor. Earlier research shows that increasing diameter, basal area and discount rate would all decrease the reservation price (Pukkala, 2006). This study confirms the earlier results that increasing tree diameter (or stand age) decreases the reservation price and increasing variation in timber price increases it (Lohmander, 1995; Gong, 1998). Increasing growth variation also increases the reservation price (Figure 5). This study also shows that it is optimal to maintain all tree species in the stand for the whole rotation when future timber prices and growth rates are uncertain; without uncertainty, it is optimal to remove the birches in the first thinnings. Increasing risk-aversion accentuates these effects. Adaptive optimizations suggest longer average rotations than anticipatory optimizations. When timber prices vary, it is beneficial to delay the clearfelling more often than to have it earlier (Lohmander, 1995; Brazee and Bulte, 2000; Gong and Yin, 2004). It is also beneficial to distribute the
OPTIMIZATION OF STAND MANAGEMENT 471 Figure 9. Distribution of the volume removed in final felling under growth and price scenario no. 7 in the optimal adaptivemanagement schedule for risk-avoider, risk-neutral or risk-taker owner. Figure 8. Stand development in the optimal adaptive-management schedule under growth and price scenario no. 7 with a riskavoider, risk-neutral or risk-seeker forest landowner. Note that the cutting years are different in different growth and timberprice scenarios. incomes (or NPVs of cuttings) more evenly among different cutting events when risk and risk-aversion increase. Our results agree with those of Penttinen (2006) who concluded that both price and growth volatility increase the optimal rotation lengths. Pukkala and Miina (1997), using stochastic multi-objective optimization, obtained the same result for a conifer mix. Our results suggest that risk-avoiders should thin earlier than risk-neutral or riskseeking decision makers; however, the rotation length is not affected by risk attitude. Gong (1998) concluded that, in the absence of thinnings, risk-avoiders should harvest earlier than risk-neutral forest landowners. Accordingly, the reservation price obtained by Gong (1998) was higher for the risk-neutral landowner. The forest-management instructions of Finland (Anonymous, 2006) are supported by deterministic optimizations. Because most forest landowners are riskavoiders and both growth and timber prices vary, our results have the following practical implications: longer rotations should be used than suggested by the instructions; mixed stands should be favoured more than suggested by deterministic calculations; cutting incomes should not be concentrated as much on the final felling as currently instructed; and it is favourable to maintain several timber assortments permanently in the stand, which means that more uneven-sized stand structures should be pursued. Rollin et al. (2005) also arrived at similar conclusions for uneven-aged management; counting for risk led to stands that were much more diverse than suggested by deterministic solutions. A mixed, uneven-aged stand allows the forest landowner to flexibly sell trees that have a good selling price and increase the share of tree species that react positively to climatic variations. When calculating the NPV for a management schedule, future rotations are assumed similar to the first one, which is not the case when there is a growth trend. If the next rotations are clearly more productive than the first one, this might have a shortening effect on the first rotation. However, this type of effect would be very small with a three per cent discount rate. Another simplification made in the calculations was that of using the multiplier of the trend line of growth as the growth-rate indicator. The forest landowner does not know this multiplier when managing the forest. Therefore, for practical adaptive management, the trend multiplier should be replaced by another type of growth index, for instance the ratio between the measured growth of some sample trees and the normal growth of the same trees, calculated using a growth model. This would most probably be a better growth-performance index than ours because it would also depend on the autocorrelated annual growth variation around the trend, which would most probably have an impact on the optimal timing of cuttings.
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