Optimal structure and development of uneven-aged Norway spruce forests

Optimal structure and development of uneven-aged Norway spruce forests Olli Tahvonen Department of Forest Sciences University of Helsinki Latokartanonkaari 7 000140 University of Helsinki Finland olli.tahvonen@helsinki.fi tel. +358504156565 fax: +358919158100

Optimal structure and development of uneven-aged Norway spruce forests Abstract Optimal harvesting of Norway spruce forests is studied applying a single-tree model for uneven-aged management. Optimization is carried out by gradient-based, large-scale interior point methods. Assuming volume maximization and natural regeneration, it is optimal to apply uneven-aged management. Allowing artificial regeneration the result is the reverse. Economically optimal solutions with a 20-year harvesting interval produce an annual sawn timber output of 4.4-2.4m 3 depending on thermal zone and interest rate. Before harvest basal area varies between 18 and 12m 2 and the diameter of harvested trees between 15 and 33 cm. In contrast with the classic inverted J-structure, optimal steadystate size structure resembles a serrate form. Profitability of even- and uneven-aged management is compared assuming the initial stand state represents an optimal unevenaged steady state. A switch to even-aged management is optimal given most favorable growth conditions and interest rate below 1-2%. In other cases, it is economically optimal to continue uneven-aged management although volume output remains lower than under even-aged management. Keywords: uneven-aged forestry, even-aged forestry, individual tree model, optimal harvesting, age-structured models, size structured models

1. Introduction Forest policy in Nordic countries is strongly oriented toward managing boreal conifer forests as even-aged stands (Siiskonen 2007). The main argument for the prevailing evenaged management system has been its economic superiority compared to selective cutting. However, scientific evidence is incomplete. To date there are no convincing studies showing that for shade tolerant species like Norway spruce, even-aged management is superior to alternatives like selective cuttings. In contrast with the present practices, a recent survey on forest policy legitimacy (Valkeapää 2009) found that a majority of Finnish citizens and large share of forest owners prefer to have alternatives in forest management and do not favor clearcuts. This study sheds some now light to this puzzle by extending the existing understanding applying a single tree model together with economic optimization. Most earlier studies such as Andreassen (1995) and Andreassen and Oyen (2002) have performed silvicultural experiments where a mature coastal Norway spruce stand has been managed applying clearcuts, single-tree selection, or group selection. According to the results, single-tree selection yields about 85% of the income obtained by clearcuts. In contrast, studies based on optimization have shown that uneven-aged Norway spruce stands may yield higher net present values of revenue compared to even-aged management (Pukkala et al. 2010; Tahvonen et al. 2010). In Wikström (2000) the result is the reverse: uneven-aged management yields about 90% of the even-aged revenues. However, none of these studies have yet applied unrestricted single-tree models together with general dynamic optimization approach. The aim of this study is to solve economically optimal structure and development for uneven-aged Norway spruce stands applying a single tree growth model, detailed biological and economic data and most general economic optimization principles. 3

Existing economic studies on uneven-aged management based on single-tree models are rare. Exceptions include Haight (1987) and Haight and Monserud (1990, 1991) who apply multiple-species, single-tree models in the western United States. Their optimization method is the Hooke and Jeeves (1961) algorithm. The present study includes differentiable functions and the optimization can be based on the most efficient gradient-based algorithms for large-scale nonlinear problems. This allows for the use of many state variables (up to 60) and long time horizons (e.g., 400 periods) without any need to simplify the single-tree model. In this study, the single-tree model determines ingrowth endogenously, in contrast to fixed ingrowth in Wikström (2000). The optimization problem is solved as a general dynamic problem, in contrast to theoretically problematic investment-efficient steady states in Pukkala et al. (2010). Dynamic optimization yields both theoretically correct steady states and paths converging toward long-run equilibria. Tahvonen et al. (2010) used transition matrix model with fixed size classes and concentrate to one thermal zone only. This study applies a more detailed single-tree model and extends the results over five thermal zones. Because of the above differences with existing literature this study presents several new results on economically optimal uneven-aged management and its profitability with respect to traditional even-aged management. It is shown that in spite of the large number of state variables and nonlinearities the volume maximization unevenaged solutions converge to steady states with constant harvest and stand structure. However, independently of terminal zones volume output is maximized applying artificial regeneration and even-aged management. In contrast, economic outcome is maximized applying uneven-aged management the exceptions being the most favorable thermal zones and interest rates below 1-2%. Economically optimal solutions converge to steady states where the tree size distributions resemble a serrate form instead to the traditional 4

inverted J-structure. In addition, results yield information on the advantages of using a more complex single tree model compared to transition matrix model with less demanding computational requirements. The paper is organized as follows. The next section introduces the optimization problem, the empirical specification used and the numerical solution methods. The results section shows first the outcomes based on volume maximization and then proceeds to economically optimal solutions. Next the economic solutions are compared to outcomes in even-aged management. Discussion section compares the results to those in the previous studies. Finally, the conclusions section presents some forest policy remarks and open questions to future studies. 2. The model and empirical specifications The optimization problem Let x,s = 1,...,n + 1, t = 0, 1, 2,... st denote the number of trees in age class s and variables d, s= 1,..., n, t = 0,1,2,... the diameter (cm) of age class s trees. Both are given in the st beginning of period t. Harvested trees are h, s= 1,..., n, t = 0,1,2,... st Natural regeneration is given by the ingrowth function φ( x ) t, d t. The fraction α s( xt, d t), s= 1,..., n of trees survives to the next period. Thus the numbers of trees evolve according to x 0 given, (1) x ( x, d ) h,t,,... (2) = 1,t 1 φ = + t t 1t 01 x = s 1,t 1 x stα s( x t, d t ) h st, s = + + 1,...,n, t = 0, 1,... (3) Note that harvest occurs at the end of each period. In (3) trees die naturally if they reach age class n and are not harvested. 5

The tree diameter in the youngest age class is δ 0 ( x t, d t ) and the diameter increment in age class s in period t is δs ( xt, d t). Thus, the tree diameter distribution evolves according to d 0 given (4) d ( x, d ), t,,... (5) = 1,t 1 δ = + 0 t t 01 d = s 1,t 1 d + st δs( x t, d t ), s = 1,...,n + + 1, t = 01,,... (6) The volumes of saw logs and pulpwood are denoted by functions ν ( d ), i= 1,2 i st 3 ( m ). Let,1,2 refer to the associate prices. The harvesting cost is C ( h, d ) and the discount p i factor b(=1/(1+r), where r is the interest rate. The problem is to t t n t max [pv(d 1 1 s + 1,t+ 1)hst + p2v 2(d s + 1,t+ 1)h st ] C( h t, d t+ 1) b (7) { h st,t = 1,...,s = 1,...,n } t = 0 s = 1 subject to (1)-(6) and the constraints h 0, x 0, = 0,1,... t t t Note that harvest occurs at the end of each period and the diameter of a tree harvested from age class s equals its level in age class s+1 in the beginning of period t+1. When the aim is to maximize timber volume production, C = 0, p = 1,i= 12,, and b=1. The cost function includes fixed cost i with the implication that it may not be optimal to harvest the stand every period. This will be taken into account by computing the model by different harvesting intervals, i.e., assuming that h = 0, s = 1,...,n for t ik, where i = 0, 1,... and k 1 is an integer. st (8) 6

Empirical specifications In their study on Finnish uneven-aged forests, Pukkala et al. (2009) estimated the ingrowth, survival, diameter growth, and height functions for Scots pine, silver birch, and Norway spruce. In this study these functions are used only for Norway spruce and Myrtillus (MT) site types. The estimated ingrowth function is φ(, ) e 4121. 0712. B t + 0083. Nt xt d t = 1, (9) where B t is the stand total basal area and N t is the total number of trees. The fractions of trees that survive over the next five year period are given as s t t { 1 4 418 1 423 1 046 0 0954 } 5 / [. +. d 6 st. lnb t. B st ] α ( d, x ) = + e, (10) where B st is the basal area of trees with a larger diameter than age class s. The function that gives the growth of the tree diameter takes the form δ 5 317 0 043 st 0 486 t 1124.. B. lnb s( dt, xt ) =. e + 0455. dst 2 0 000927 st 0 18 + 0 823. d.. lnt, sum (11) where Tsum is the temperature sum in degree days (d.d.). It varies between 900 and 1,300 d.d. The diameter of the youngest trees is given by 2. 004 0. 101 lnb t 0. 0176 d( 0 xt,d) t = e. (12) 7

The height of trees is a function of diameter: 39. 691 h(d e st ) =. 2 1 + 25. 683 / d st + 37. 785 / dst (13) The functions for saw timber and pulpwood volumes are obtained using Heinonen (1994) and equation (13). The maximum and minimum lengths for saw logs are 5.5 and 4.3 meters, respectively. The minimum diameters for saw logs is 15cm and for pulp logs 6 cm. In order to maintain smooth model properties the volumes (m 3 ) are given as: 1 2 3 2 3 ( st st st ) ν = 10 116. 0906 31. 1854 d + 1. 9407 d 0. 0121 d, (. d st. d st. d st. ) ν = + 3 3 2 10 0 0068176 0 660699 18 2853 72 8905, (14a,b) where 12 40. The roadside price for saw logs is 52 and for pulpwood 28. d st Harvesting costs are from Kuitto et al. (1994). These detailed models (Appendix A) were originally estimated for even-aged management. They contain separate models for cutting and hauling costs and for thinnings and clearcuts. Following Surakka and Siren (2004), the cost model for uneven-aged management is formed by taking the hauling cost from the thinning model and the logging costs from the clearcut model and multiplying the latter by a factor of 1.15. The fixed harvesting cost equals 300. Optimization procedure Taking into account the empirical specification (9)-(14b), the optimization problem (1)- (6) is smooth in the sense that all functions are continuous and differentiable. Thus it is 8

possible to use efficient gradient-based optimization algorithms that are built on the Karush-Kuhn-Tucker theorem of nonlinear programming. This study applies Knitro optimization software that includes state-of-the-art interior point algorithms (Byrd et al. 2006). The difficulties follow from the rather large number of state variables (40-60) and nonconvexities, i.e. the possibility of several locally optimal solutions. To find the global optima it is necessary to use 15-100 randomly chosen initial guesses until further guesses do not increase the objective function value. 3. Results Maximizing volume yield It is useful to first study the model properties in the simplest (theoretical) case where trees are harvested every period (i.e., every five years) and the aim is to maximize volume yield or the saw timber volume yield. In this setting, with no discounting, it is possible to find the optimal steady state by solving a static optimization problem, i.e., {h s,s = 1,...,n} s = 1 1 1 s+ 1 s s s 1 0 n max [ v ( d ) + p v ( d )] h x = φ( xd, ) h, s+ 1 s s 1 s+ 1 2 2 s+ 1 s x = x α ( xd, ) h, s = 1,...,n, d = δ ( xd, ), d = d + δ ( xd, ), s = 1,...,n 1. (15) For this computation 20-30 age classes are needed to guarantee that the solution is not restricted by the number of age classes. The results are given in Table 1. Both total and saw timber yield increase while the basal area and number of standing and harvested trees tends to decrease with the temperature sum. The explanation is presented in the next two lines: given the highest temperature sum, it takes 60 years after ingrowth until tree 9

diameter is about 29 cm and cutting is optimal. At the lowest temperature sum, the diameter of harvested trees is about 23 cm when tree age is 90 years after ingrowth. Thus, when tree growth is slower, the forest includes a higher number of age classes and trees compared to when tree growth is faster. Although the basal area becomes higher, the higher number of trees implies that ingrowth is higher with lower temperature sum (cf. Equation 9). Comparing ingrowth with the numbers of harvested trees shows that with these stand densities only 0-2 trees per cohort die. In all cases it is optimal to cut all trees from the oldest age class and leave others untouched. Table 1. Optimal steady states for maximizing total yield Table 2. Optimal steady states for maximizing saw timber yield Table 3. Maximizing total yield under even-aged management Figures 1a and b. Optimal stand structure after harvest Figures 2a and b. Steady-state total yield and the productivity of standing volume Table 2 shows similar results when the aim is to maximize annual saw timber yield. Total yield is now lower, standing volume increases, trees are cut later with larger diameters and the numbers of harvested trees are lower. Together the results in Tables 1 and 2 suggest that the model output is coherent. Figures 1a and b show the stand age structure at the steady state. In Figure 1a, the number of trees decreases with size. This is in line with the classical inverted J-curve model The stand structure follows from natural mortality and the fact that ingrowth remains constant given constant harvest. Figure 1b shows the steady-state tree diameter pattern and, equivalently, how tree diameters grow until they are cut. Another view on the model properties and on the production capacity of the uneven-aged Norway spruce stand can be obtained by computing the steady state surplus production. This is the periodic yield net of natural mortality as a function of standing 10

Table 1. Optimal steady states for maximizing total yield T sum 1300 1200 1100 1000 900 annual total yield,m 3 5.1 4.9 4.7 4.5 4.3 annual saw timber yield m 3 4.5 4.3 4.0 3.8 3.4 basal area before cut m 2 12.3 13.4 13.4 14.7 15.8 basal area after cut m 2 9.6 10.8 10.9 12.2 13.3 no. of standing trees before cut 552 627 708 848 1017 no of harvested trees a -1 8.4 8.3 9.3 9.8 11.6 standing vol. before cut m 3 98 106 103 109 111 age * of harvested trees, years 60 75 75 85 90 diameter of harvested trees cm 28.7 28.4 26.5 26.5 23.3 ingrowth, no. of trees a -1. 8.7 8.6 9.8 10.5 12.7 Note: Harvest every five years, * age after ingrowth Table 2. Optimal steady states for maximizing saw timber yield T sum 1300 1200 1100 1000 900 annual total yield,m 3 5.0 4.8 4.6 4.4 4.1 annual saw timber yield m 3 4.6 4.4 4.1 3.9 3.6 basal area before cut m 2 14.0 15.0 15.9 16.9 18.5 basal area after cut m 2 11.4 12.5 13.5 14.5 16.3 no. of standing trees before cut 570 639 727 848 992 no of harvested trees a -1 7.0 7.0 7.1 7.5 7.3 standing vol. before cut m 3 115 121 127 131 141 age * of harvested trees, years 80 90 100 110 130 diameter of harvested trees cm 30.9 30.4 29.6 28.4 27.8 ingrowth, no. of trees a -1. 7.4 7.4 7.8 8.5 8.9 Note: Harvest every five years, * age after ingrowth Table 3. Maximizing total yield under even-aged management temperature sum, d.d. 1300 1100 900 initial diameters for 3 size classes, cm 5.6,7.1,8.8 4.6, 6.1,7.7 4.6, 5.2,6.3 saw timber yield m 3 a -1 6.7 5.8 4.9 total yield m 3 a -1 7.6 6.6 5.6 11

Number of trees per age class 48 46 44 42 40 38 36 34 5 10 15 20 25 30 Diameter, cm 35 30 25 20 15 10 5 0 0 25 50 75 100 125 150 Diameter cm Age years 1300 d.d 1200 d.d 1100 d.d. 1000 d.d. 900 d.d. Figures 1a and b. Optimal steady state structure after harvest Objective: saw timber volume maximization, Harvest every 5 periods Total steady state yield per 5 years 28 26 24 22 20 18 16 14 12 10 50 100 150 200 Fraction of volme harvested per five years 0.6 0.5 0.4 0.3 0.2 0.1 0.0 50 100 150 200 Standing volume before harvesting Standing volume before harvesting Temperature zone (d.d.) 1300 1100 900 Figure 2a and b. Steady state surplus production and the productivity of standing volume 12

tree volume. This function is obtained by maximizing yield and varying a minimum constraint on standing volume. The outcomes are given in Figures 2a and b. In the case of 1300 d.d., the highest total yield is obtained when the standing volume is about 98m 3 before cutting. At this steady state about 42 trees are harvested every five years. Ingrowth per five years is about 43.5 trees, i.e. natural mortality decreases the surplus production of each cohort by 1.5 trees. In comparison, when the standing volume before the cut is kept at 152m 3 the yield per five years is 22.2m 3. This is about 87% of the maximized yield. Three trees are harvested from age class 120, and 19 trees from age class 125. Their diameters are 35 cm and 37 cm respectively. Ingrowth is 25 trees per five years, i.e., three trees from each cohort die. Figure 2b gives a rough view on the productivity of standing volume in fiveyear terms. For the 1300 d.d. temperature zone, it varies between 56% and 9%, implying an annual productivity between 9% and 2%. Because of the model s complexity, it is not evident that the optimal solutions converge toward steady states representing uneven-aged management with constant variables over time. It this model even-aged management is an admissible solution type instead of uneven-aged management (Tahvonen 2009). This question is studied in Figures 3a, b, and c. The optimal solution is computed from seven different initial states. In spite of the rather strong initial variation in periodic harvesting, basal area, and number of trees, all the optimal solutions converge toward the same steady-state solution. This steady state represents uneven-aged management and equals that shown in Table 1, column 3. Figures 3a, b, and c. Volume maximization over time The question of constant versus cyclical harvesting can be further analyzed by computing optimal steady-state solutions under the constraint that harvesting is only 13

100 Total yield per 5 yrs 80 60 40 20 0 30 Number of trees after cuts 800 700 600 500 400 300 25 20 15 10 5 0 10 20 30 40 50 Time in 5 yrs Basal area before cut m 2 Figure 3a,b and c. Volume maximization over time Cutting every five years, Temperature sum 1200 d.d. 14

Average annual yield m 3 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6 0 2 4 6 8 10 12 Average annual ingrowth 13 12 11 10 9 8 7 6 5 0 2 4 6 8 10 12 Harvesting interval in 5 yrs Harvesting interval in 5 yrs 900 d.d 1000 d.d 1100 d.d 1200 d.d 1300 d.d Figure 4a and b. Maximized volume yield and ingrowth as functions of harvesting interval. Revenues per 5 yrs 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 10 20 30 Time Figure 5. Convergence toward the steady state from various initial states T sum =1200, interest rate 3%. 15

Table 4. Optimal steady state solutions for maximizing present value of roadside revenues net of harvesting cost Interest rate, % T sum d.d. Total yield m 3 a -1 Saw timber yield m 3 a -1 Basal area before/after harvest, m 2 Number of trees before/after harvest Diameter range of harvested trees, cm Net revenues (20a) -1 Present value, infinite horizon 1 900 4.20 3.30 18.4/8.7 1015/772 20.9-26.1 3162 17522 2 900 3.75 2.90 15.6/6.0 874/602 19.0-25.0 2936 8978 3 900 3.75 2.58 12.9/3.8 716/428 16.7-24.0 2631 5895 4 900 3.59 2.42 12.5/3.0 650/357 15.4-23.9 2479 4560 1 1000 4.40 3.60 17.6/7.8 823/615 21.7-28.2 3421 18958 2 1000 4.22 3.27 14.95/5.18 723/481 18.4-26.9 3179 9720 3 1000 4.03 2.99 13.22/3.71 644/384 17.9-25.9 2952 6614 4 1000 4.03 2.66 11.33/2.33 542/271 15.1-25.9 2651 4877 1 1100 4.58 3.94 17.63/7.62 714/534 22.7-30.8 3648 20216 2 1100 4.47 3.64 15.15/5.08 647/431 19.7-29.0 3464 10592 3 1100 4.30 3.37 13.51/3.63 587/350 18.9-27.8 3247 7276 4 1100 4.02 3.02 11.71/2.28 506/253 15.8-27.7 2948 5423 1 1200 4.76 4.19 17.35/7.10 629/462 23.4-32.3 3845 21307 2 1200 4.70 3.97 15.38/5.01 588/391 22.9-30.9 3720 11375 3 1200 4.54 3.71 13.81/3.58 541/323 19.9-29.5 3519 7884 4 1200 4.41 3.52 12.87/2.8 507/287 18.0-29.3 3364 6188 1 1300 4.92 4.38 16.8/6.2 564/401 25.0-33.4 4019 22271 2 1300 4.88 4.21 15.9/4.5 529/341 21.3-31.7 3917 11978 3 1300 4.65 3.84 14.2/2.8 475/259 21.2-31.4 3620 8109 4 1300 4.54 3.68 13.76/2.2 448/224 18.4-30.7 3492 6424 Notes: Harvesting interval 20 years (four 5 year periods). 16

allowed to be positive every k periods (cf. Equation 8). In addition, it must be required that the steady-state stand structures (number of trees and tree diameters) are repeated at the end of the harvesting interval after each cutting. The results in Figures 4a and b reveal that both the average annual output and ingrowth decrease if harvesting does not take place every period. This is in line with the dynamic solutions in Figures 3a-c and supports the result that when maximizing volume yield and relying on natural regeneration, it is optimal to harvest trees every period. Figure 4. Maximized volume yield and ingrowth as functions of harvesting interval. Table 4. Maximizing total yield under even-aged management Ideally it should be possible to use the same forest growth model for both unevenand even-aged management. Results from such an experiment are given in Table 3. The second row shows the thermal zone-specific initial states (25 years after clearcut) assuming that the initial stand is represented by three size classes. In all cases, the initial number of trees is 1700. Of these, 24.5% is in the largest size class, 49% in the middle size class, and the rest is in the smallest size class. In maximizing total yield, the sizespecific intensity of two thinnings and their timing are optimized together with the rotation period. Comparison with Table 1 suggests that artificial thinning with even-aged management is capable of producing higher volume yield than uneven-aged management based on natural regeneration. Maximizing present value of net forestry income Maximizing economic net revenues brings the price difference between saw timber and pulpwood, the harvesting costs, and the interest rate into the computations. Because harvesting costs include a fixed cost component equal to 300, it is not optimal to harvest the stand every period. For simplicity, the results will be computed assuming a 17

cutting interval equal to 4 periods (20 years). With a low interest rate (1%) a harvesting 8000 Total revenues, Basal area before harvest Number of trees after cut Sawn timber yield per 5 yrs, m3 Harvest, number of trees per 5 yrs 7000 6000 5000 4000 3000 20 15 10 5 0 600 500 400 300 200 100 140 120 100 80 60 40 20 0 70 60 50 40 30 20 10 0 20 40 60 80 100 Time in 5 yrs periods Revenues net of harvesting costs, Basal area after harvest Harvested trees Pulp yield per 5 yrs, m3 Ingrowth, number of trees per 5 yrs Figure 6. Optimal solution for maximizing roadside revenues net of harvesting cost Note: T sum =1200, interest 1% 18

interval equal to 5 periods may yield a slightly higher net present value of revenues, as will an interval equal to 3 periods in the case of a high interest rate. Figure 5 shows periodic net revenues from seven optimal solutions starting from different initial stand states. They all converge toward the same steady-state equilibrium. Figure 6 shows a more detailed example for an optimal convergence toward the steadystate stand structure and harvesting. The basal area, number of trees and ingrowth fluctuate because harvests only occur once every five periods. For the same reason ingrowth is highest at the period following low basal area and low number of trees. Figure 5. Convergence toward the steady state from various initial states Figure 6. Dynamic solution for maximizing roadside revenues net of harvesting cost over time Computing the optimal dynamic solutions 1 for thermal zones between 1300-900 d.d. and for interest rates between 1-4% produces the steady states characterized in Table 4. Given any interest rate, a higher terminal zone always yields higher total and saw timber yield as well as higher steady-state net revenues. Similarly, as in maximizing volume yield, optimal steady-state stand density decreases with thermal zone. Increasing the interest rate decreases saw timber yield and periodic revenues but may increase total yield (and pulpwood). In addition, a higher interest rate implies lower steady-state density as measured by basal area or number of trees. However, when stand density decreases, trees grow faster in diameter. Thus, compared to steady states for volume output maximization, the decrease in saw timber yield is quite low. With a 1% interest rate the optimal solutions produce about 95-90% of the maximum volume output and with 4% the output is about 80-70% of the maximum. The volume obtained per harvest varies between 76m 3 and 105m 3. The harvesting cost level (not shown) varies between 8.7 /m 3 and 7.4 /m 3. The figures for the basal area and number of trees do not contain trees below the ingrowth diameter. This basal area may be expected to vary between 1-2m 2. Table 4. Optimal steady states for maximizing present value of roadside revenues 19

(a) (b) 100 100 Number of trees per size class 80 60 40 20 Number of trees per size class 80 60 40 20 0 5 10 15 20 25 Diameter cm 0 5 10 15 20 25 Diameter cm Number of trees per size class 70 60 50 40 30 20 10 0 (c) 5 10 15 20 25 30 Number of trees per size class 100 80 60 40 20 0 (d) 5 10 15 20 25 30 Diameter cm Diameter cm 60 (e) 100 (f) Number of trees per size class 50 40 30 20 10 Number of trees per size class 80 60 40 20 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 Diameter cm Diameter cm Remaining trees Harvested trees Figure 7. Optimal tree structure at the steady state a) T sum =900, r=0.01, b) T sum =900, r=4% c) T sum =1100, r=0.01, d) T sum =1100, r=4% e) T sum =1300, r=0.01, f) T sum =1300, r=0.04% 20

Number of trees 180 160 140 120 100 80 60 40 20 0 (a) 5-10 10-15 15-20 20-25 25-30 30-35 Diameter classes, cm Number of trees 160 140 120 100 80 60 40 20 0 (b) 5-10 10-15 15-20 20-25 25-30 30-35 Diameter classes, cm Remaining trees Harvested trees Figure 8. Stand structure in 5 cm size classes 1300 d.d. a) r=1%, b( r=2% 21

Figure 7a-f. Optimal tree structure at the steady state Figures 7a-f show steady-state stand structures and harvested trees. Harvesting removes the four largest age classes while others are left growing. When the number of trees in the (endogenously-determined) size classes is depicted, as in Figures 7a-f, the stand structures represent a serrate form. This structure is determined by the 20-year harvesting interval and the associated variation in stand density and ingrowth. After each harvest, stand density is low, implying that ingrowth is high. During the next four periods density increases, implying lower ingrowth and smaller tree cohorts. Figures 7a-f show a mirror image of this process because cohort age increases along the x-axes. Figure 8. Stand structure in 5 cm size classes 1300 d.d. a) r=1%, b) r=2%. To compare this structure with the traditional inverted -J structure in the unevenaged literature, it must be noticed that in Figures 7a-f the frequency of the endogenously determined size classes decreases with diameter. Figures 8a and b depict the steady states of 1300 d.d. for interest rates of 1% and 4% when trees are grouped into 5 cm diameter classes. The number of trees becomes highest in the smallest size class, but in general the size-class structure still clearly differs from the classic inverted J-curve. Comparing Figures 7e and 7f to 8a and 8b reveals that the number of trees tends to decrease with size not because of natural mortality but because the diameter differences between size classes increase in time and size. Comparing the economic performance of uneven- and even-aged management Comparing the economic outcomes from even- and uneven-aged management requires information on maximized bare land value under even-aged management. One possibility is to take the bare land values from existing studies. The advantage of this choice is that several existing studies apply stand growth models that have been designed for even-aged 22

Table 5. Comparing the even-and uneven-aged management T sum, d.d. Interest rate, % BLV this study Total yield m 3 a -1 BLV Pukkala et al. 2010 BLV Hyytiäinen et al. 2010 clearcut value Break even BLV 900 1 11092 5.4 10360 2560 4828 12694 900 2 2378 4.5 2146-55 4004 4974 900 3 285 4.2 353-641 3189 1369 1100 1 14779 6.3 15057 9875 5540 14676 1100 2 3800 5.4 4532 2335 4514 6078 1100 3 1017 5.2 1583 403 3887 3389 1300 1 18537 7.4 20673 20448 5639 16632 1300 2 5230 6.4 7800 6099 4945 7033 1300 3 1778 6.2 3695 2208 4122 3987 Table 6. Investment efficient steady states T s, d.d. Interest rate % NPV* NR T BLV** NPV Pukkala et al. 2010 900 1 17066 4035 11092 15283 2 7185 3863 2378 6180 3 4082 3627 285 3371 1100 1 19536 4523 14779 18082 2 8397 4464 3800 7432 3 4832 4215 1017 4217 1300 1 21669 4981 18375 19954 2 9393 4890 5230 8504 3 5462 4789 1778 4937 *Investment efficient steady state without requiring the Weibull distribution **BLV computed in this study 23

management and should offer reliable results. In addition, these studies describe stand growth based on artificially planted seedlings and thus include the potential genetic superiority compared to natural regeneration. The drawback is that various details like prices and harvesting costs may deviate from the values used here. Another possibility is to compute the bare land values applying the specifications (9)-(14) and exactly comparable economic parameter values. Here, both of these approaches are used. To compute the even-aged solution requires solving for the number and timing of thinnings, the number of harvested trees in thinnings, and the rotation period. The initial state is taken to be the same as in Table 3. Regeneration costs are given as 2 11 W(N,r) = +. b + b, 0 425 1700 0 39 292 where the first cost item refers to clearing and mounding the regeneration area, the next to planting costs (1700 seedlings) that materialize two years after the clearcut, and the last to tending costs that materialize 11 years after the clearcut. Harvesting costs are taken from Kuitto et al (1994). The optimized number of thinnings varies between zero and two. Table. 5. Comparing even- and uneven-aged management The results for maximizing bare land values are shown in Table 5, column 3. Column 4 shows the associated average annual volume output per hectare. Bare land values are somewhat lower than in Pukkala et al. (2010) where timber prices were higher and regeneration costs lower. The differences from the bare land values of Hyytiäinen et al. (2010) may follow from lower regeneration costs and considerably lower stand growth in the 900 d.d. thermal zone. To compare the economic superiority of even- and uneven-aged forestry, the task is to solve the problem 24

T n t T max [ p 1ν 1( d s,t+ 1) + p2v 2( d s,t+ 1)]hst C t ( h 01 1 i t, d t ) b + b V * (16) { h st,t =,,...,T, s =,...,n,t } t = 0 s = 1 subject to restrictions (1)-(6), where V* is the maximized bare land value. The essential feature in this problem is the optimal choice of a possible switch from uneven-aged management to even-aged management. The choice T = implies that uneven-aged management is applied forever; choosing some finite T 0 implies a switch to even-aged management. Complete analysis of (16) is tedious since the optimal T may depend on the initial state, and the set of possible initial states is infinite. A computationally simple possibility is to assume that the initial state equals the optimal uneven-aged steady state at the beginning of the period with harvest. Given this initial state the simplest possibility is to compare the cases T = and T = 0, i.e., the solution of following uneven-aged management forever and the solution where the stand is clearcut immediately and managed as an even-aged stand thereafter. Let R UEA denote the steady state net revenue th (from Table 4) that is obtained every k period under optimal uneven-aged management. In addition, let R cc denote the net revenues from a clearcut of this same stand (applying the clearcut harvesting cost function (Kuitto et al 1994). An immediate switch to evenaged management is at least as profitable as continuing uneven-aged management if R 1 b UEA 5k R + V*. cc (17) When (17) is an equality, V* may be defined as the break-even bare land value. Table 6. Stand clearcut value and break even bare land value 25

Present values, 20000 18000 16000 10000 8000 6000 4000 2000 (a) 0 20 40 60 80 100 120 Present values, 20500 20000 19500 19000 10000 8000 6000 4000 (b) 0 20 40 60 80 100 120 (c) Present values, 24000 22000 12000 10000 8000 6000 0 20 40 60 80 100 120 Transition period, in 5 yrs Interest rate=0.01 Interest rate=0.02 Interest rate=0.03 The present value from uneven-aged management Figure 9. Uneven-aged management compared to switches to even-aged management a) 900d.d. b) 1100 d.d c) 1300 d.d. 26

The results for equation (17) are given in Table 5, column 8. The bare land values obtained in this study exceed the break-even bare land values in two cases: when the temperature sum is 1100 d.d or 1300 d.d. and the interest rate is 1%. In addition, the bare land values in the study by Pukkala et al. (2010) exceed the breakeven level when the temperature sum is 1300 d.d. and the interest rate is 2%. The study by Hyytiäinen et al. (2010) yields higher bare land values in one case: when the temperature sum is 1300 d.d. and interest rate is 1%. These results are intuitive: even-aged management becomes competitive with a lower interest because the interest costs of artificial regeneration decrease. In addition, stand growth is faster with higher temperature sum and costly artificial regeneration becomes profitable compared to free but scanty natural regeneration. Figure 9. Uneven-aged management compared to switches to even-aged management. Figures 9a-c shows results on possible later switches to even-aged management. The figures have been computed using the bare land values obtained in this study. Thus, in line with Tables 5 and 6, the immediate switch to even-aged management is optimal given the temperature sum is either 1100 d.d or 1300 d.d., and the interest rate equals 1%. Finally, it must be emphasized that this comparison allows thinning from above in evenaged management. If only thinning from below were allowed, the bare land values would be lower than the break-even bare land values in all cases. 5. Discussion This study applies a single-tree model for analyzing the economics of uneven-aged management. The ecological model has been applied without any simplifications or ad hoc constraints. The objective function includes detailed, empirically-estimated harvesting cost functions specified for the purposes of uneven-aged management. The 27

optimization problems (with discounting) are solved in their most general form, i.e. by computing finite time solutions with a long enough planning horizon to obtain a close approximation of infinite horizon solutions. The most important findings of the study are: 1. Maximization of volume output and reliance on natural regeneration yields uneven-aged management as the optimal solution. 2. When the thermal zone varies between 900 and 1300 d.d., volume output varies 3 1 1 between 5.1 and 4.3 ma ha, the diameter of harvested trees between 23 and 29 2 cm, and the pre-harvest basal area between 16 and 12 m. 3. The dynamically optimal solutions converge toward the optimal uneven-aged steady states independently on the initial stand structure. 4. Even-aged management and artificial regeneration (1700 seedlings) yield higher volume output than uneven-aged management when both solutions are computed by the same empirical single-tree growth model. 5. When the thermal zone varies between 900 and 1300 d.d. and the interest rate varies between 1% and 4%, the economically optimal output varies between 4.9 and 3.6 ma ha 3 1 1, the diameter of harvested trees between 15-33 cm and the pre- 2 harvest basal area between 18 and 13 m. 6. In economically optimal solutions, the steady-state size class structure resembles serrate form instead of the traditional inverted J-curve. 7. Given the optimal uneven-aged steady state as the initial state, it is optimal to switch to even-aged management only if the interest rate is 1%-2% or lower and the thermal zone 1100 d.d. or higher. 28

Until now, similar economic optimization based on a single-tree model has not been performed for Norway spruce. However, although the model specifications differ, it is possible to compare these results with earlier uneven-aged studies on Norway spruce. Among the first attempts to specify a growth model for uneven-aged Norway spruce is the (nonlinear) transition matrix model by Kolström (1993). In his model, ingrowth depends on the gaps left by harvested trees (cf. Usher 1966). Assuming harvest removes some fixed proportion of trees from each size class, Kolström shows that the uneven-aged stand may produce up to 8m 3 ha -1 a -1 (1200 d.d. sites between high and medium fertility). When this model is applied (Tahvonen 2007, 2009) to maximize present value stumpage revenues (3% interest rate, harvest every 5 years) the steady-state volume output equals 7m 3 a -1 ha -1. The diameter of harvested trees is 26 cm and the afterharvest basal area 17m 2. In stylized comparison, the present value of stumpage revenues from uneven-aged management was clearly higher than that of even-aged management. It is important to recognize that Kolström s (1993) specification yields excessive ingrowth, and then it becomes optimal to thin the smallest size class rather heavily. In addition, the transition from the smallest size class to the next is density independent. This explains the high basal area levels compared to the results here. Andreassen (1995) reports results on 16 long term experimental plots and compared the volume output between even-and uneven-aged management. According to his results uneven-aged management yields about 15-20% lower volume production. When compared to the results based on artificial planting and systematic volume maximization (Tables 1 and 3) these losses here are somewhat higher (25-32%). Wikström (2000) uses six separate studies in constructing a single-tree model for Norway spruce. He refers to Kolström (1993) and specifies ingrowth independently of stand density, applying fixed ingrowth equal to 50 trees per five years with an average 29

diameter equal to 5 cm. To reduce the number of decision variables, cuttings are specified within 6 cm size classes. Post-harvest standing volume is required to be at least 150m 3. Harvesting cost is independent of the management method. The interest rate equals 3%. According to the results, the optimal even-aged solution yields an average volume output of 3 1 1 3 1 1 63.ma ha while uneven-aged management yields 3. 2m a ha. Basal area remains above 16m 3 and trees are cut when they reach 19-25 cm in diameter. The optimal uneven-aged solution is computed over 41 (5-year) periods and yields 96-88% of the net present value of revenues for the even-aged solution. When these results are compared with the results of the study at hand it is 3 essential to note that a lower bound constraint on standing volume (150 mha 1 ) must decrease the economic outcome from uneven-aged management (cf. Tables 1, 2, and 5, and Figure 2a). Even when this restriction is taken into account, the volume yield from uneven-aged management is low compared to the results in this study. Wikström explains the low yield by some special properties of the growth model used. Pukkala et al. (2010) utilize the same growth data (i.e., Pukkala et al. 2009) as is used here. Their aim is to present practical instructions for uneven-aged management of Norway spruce and Scots pine. The economic approach is based on Duerr and Bond (1952) and Bare and Opalach (1987). Thus they search post-thinning, steady-state stand structures that maximize net present value (NPV) defined as NRT NPV = ( NR T cc NR pc ), ( 1+ r) 1 (18) where NR T is the roadside revenue net of harvesting cost (or stumpage revenues) realized every T periods, NR is the net revenue if the steady-state stand is clearcut and NR is cc the net revenue if it is cut to the steady-state, post-harvesting structure. The interpretation of this investment efficient steady state in Getz and Haight (1989: 269-272), Pukkala et pc 30

al. (2010), and others is that NR cc NR pc is viewed as an opportunity cost or investment cost of stocking needed to perform uneven-aged forestry 2. Maximizing (18) requires solving for the number of trees and their diameters over the cycle periods taking (1)-(6) as constraints. To simplify this problem, Pukkala et al. (2010) follow Bare and Opalach (1987) and assume that the post-harvesting stand structure is represented by a Weibull distribution function. The problem is computed by optimizing the parameters of the Weibull function instead of the number of trees directly. Their method yields a steady-state stand structure where the number of trees decreases with diameter and trees larger than 19 cm are cut. Typically, about 50% of the removal is pulpwood. In comparing even- and uneven-aged management the latter turns out to be superior both for Scots pine and Norway spruce, the only exception being a Norway spruce medium fertility site, 1300 d.d. thermal zone, and 1% interest rate. The study by Pukkala et al. (2010) includes two separate features that deserve attention: the investment-efficient steady state, and the strategy of optimizing the parameters of the Weibull distribution function. The investment-efficient steady state is analyzed by Getz and Haight (1989, p. 269-272), where it is shown to lack of a sound theoretical basis. One main problem is the ad hoc forest valuation represented by NR cc NR pc. As a consequence, both the steady-state stand structure and the economic profitability results become questionable. This study shows that, in general, the steady-state stand structure deviates from the classic, inverse J-curve. Solving for the post-harvesting stand structure by optimizing the Weibull distribution function parameters may not yield a stand structure depicted in Figures 7a-f or 8a and b. Thus, this solution method produces an ad hoc constraint in optimization and must decrease the economic outcome from uneven-aged management. In addition, the use of the Weibull function may explain the high fraction of pulpwood 31

obtained by Pukkala et al. (2010), that is, 50% vs. 20-30% here. The remaining question is how the investment-efficient steady state defined by (18) but computed without the Weibull simplification distorts the profitability figures. The results for this question are shown in Table 6. The cutting interval is four periods as in the comparisons by Pukkala et al. (2010). The revenues per 20 years (NR T ) are considerably higher than the optimal steady-state revenues obtained here 3 (see Table 4, column 8). In addition, the investment-efficient NPVs (third column) clearly exceed the maximized bare land values in all cases (given any estimate and independent of the interest rate and thermal zone). This shows that the investment-efficient steady-state revenues and their comparison with the bare land value will overestimate the relative economic performance of uneven-aged management. In addition, comparing the third and the last columns in Table 6 shows that simplifying the optimization using the Weibull distribution function decreases the NPVs obtained without this simplification. Table 6. Investment efficient steady states The study by Tahvonen et al. (2010) applies a nonlinear transition matrix model for studying uneven-aged management of Norway spruce. The harvesting costs and other economic features are approximately the same as they are here. Getz and Haight (1989, p. 250) compare the performance of a single-tree model and a stage-structured model very similar to the one used in Tahvonen et al. (2010). They find that both models yield very similar projections, assuming exogenously given harvesting. Here, it is possible to compare the optimal solutions obtained by these two types of models. The model in Tahvonen et al. (2010) is suitable for only one site type that is between Oxalis-Myrtillus and Myrtillus. The temperature sum is 1200 d.d. Thus, the results of this earlier study may be approximately compared to the results obtained assuming 1300 d.d. in this study. Their volume maximization results are very similar to 32

the results here, the only difference being that the transition matrix model yields slightly larger tree diameters and volumes for harvested trees. Similarly as here artificial regeneration with thinning from above produces higher volume than uneven-aged management although the latter dominates assuming natural regeneration. Economically optimal solutions can be compared assuming a 3% interest rate. Tahvonen et al. (2010) used a 12-year harvesting interval. Before- and after-harvest steady-state basal area levels were 10 and 4m 2, total and saw timber yield 4.6 and 4.1m 3 a - 1 ha -1, and revenues at an annual level 178 ha -1. At the steady state the diameter of harvested trees was between 23 and 39 cm. Again, the results are rather close to results in Table 4. However, since the results in Table 4 are based on a 20-year harvesting interval, the basal area and the number of trees reach higher levels before the harvest. The other difference is that in Table 4 the diameter of harvested trees is lower, implying lower saw timber yield. In addition, the diameter variation in the transition matrix model is larger than the diameter variation in the single-tree model (16 vs. 10 cm) although in the later case the harvesting interval is longer. This difference follows from the fact that the transition matrix model yields too slow growth for some trees and too fast for others. In Tahvonen et al. (2010), a harvesting interval of 12-15 years yields similarly fluctuating ingrowth as here. However, at the steady state the number of trees is monotonically decreasing, although with diameters between 15 cm and 27 cm, their number remains almost constant. The transition matrix model is not capable of producing a serrate stand structure because only a fraction of trees (depending on the stand density) moves to the next size class, implying that the variation in ingrowth is smoothed out. In contrast the single-tree model used here maintains the original cohorts and variation in ingrowth. 33

In comparing the economic superiority of forest management forms, Tahvonen et al. (2010) found that given a 3% interest rate, the break-even bare land value is 4450. This is somewhat higher than here (3989 ) and may be a consequence of the transition matrix model that overestimates the volumes of harvested trees. Overall, Tahvonen et al. (2010) found that uneven-aged management was superior to even-aged management, although with low interest rates the difference becomes small. In comparison, the study at hand shows that the relative superiority of even-aged management increases with thermal zone. An obvious explanation is that when forest growth is faster, the interest costs related to artificial regeneration obtain faster payoff. 6 Conclusions Finnish forest policy has strongly supported even-aged management and forest legislation is designed to prevent selective cuttings. At the statute level, forest harvesting operations are classified into two groups: stand improvement cuttings and regeneration cuttings. For stand improvement cuttings, the forest statute (528/2006) specifies the lower bound basal area level that must be met after each stand improvement cutting. The lower bounds depend on the dominant height of the stand, and are site fertility and thermal zone specific. If these restrictions are applied to the 1% interest rate cases and to the thermal zones of 900 d.d., 1100 d.d., and 1300 d.d. in Table 4, the lower bounds for the basal areas after cuttings become 10, 13, and 13m 2. In comparison, the corresponding after-cut basal areas in Table 4 are 8.7, 7.6, and 6.2m 2. With higher interest rates, the after-cut basal areas become lower, implying that all the solutions in Table 4 violate the legal lower bound restrictions. By requiring higher than optimal stand density the restrictions decrease ingrowth, and future harvesting possibilities. In addition, the statute states that in stand improvement cuttings the residual trees must be primarily the trees from the 34

highest canopy layers. It is somewhat unclear how this statement should be understood, but straightforward interpretation suggests that the steady-state cuttings described in Figure 7a-f violate this statement. It has been shown elsewhere (Hyytiäinen and Tahvonen 2003) that in the case of even-aged management Finnish forest policy promotes maximum sustainable yield. The results of this study suggest that strong promotion of even-aged management is in line with this policy and that uneven-aged management may yield about 15-25% less timber. The obvious question is why volume maximization is the primary goal of forestry and not economic surplus. In addition to the direct losses to forest owners, it tends to decrease the market price of timber, causing another income distribution effect between forest owners and forest industry. Future economic studies on uneven-aged management would greatly benefit from further developments in growth and yield models. In particular, the ingrowth specification should describe the earlier development of seedlings instead of implicitly assuming the existence of a pre-ingrowth seedling stock independent of stand state, as in the present specification. Economic models should also be extended to analyze mixed species stands including Norway spruce, Scots pine, and silver birch (cf. Bollandsås et al. 2008). Such extension will be necessary to increase understanding of the potential to apply unevenaged management in Nordic conditions. Notes: 1. The dynamic solutions are computed using time horizons from 100 to 350 periods depending on the interest rate. These horizon lengths are long enough to produce a close approximation toward the optimal steady state. 35