An Optimal Management Strategy of Carbon Forestry with a Stochastic Price

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1 sustinbility Article An Optiml Mngement Strtegy of Crbon Forestry with Stochstic Price Sor Yoo 1, Yong-sung Cho 2 nd Hojeong Prk 2, * 1 Estimtes nd Tx Anlysis Deprtment, Ntionl Assembly Budget Office, Seoul 07233, Kore; sorh604@gmil.com 2 Deprtment of Food nd Resource Economics, Kore University, Seoul 02841, Kore; yscho@kore.c.kr * Correspondence: hjeongprk@kore.c.kr; Tel.: Received: 16 August 2018; Accepted: 10 September 2018; Published: 14 September 2018 Abstrct: An nlysis for the vlue of crbon forestry needs to be provided for the successful estblishment of the crbon offset mrket in Kore. We present n optiml mngement strtegy for forest owner who prticiptes in the offset mrket. Given stochstic process of the timber price following geometric Brownin motion, the profit mximiztion problem of the forest owner is solved. The model finds n optiml hrvest time in the presence of the crbon nd timber revenues with opposing time effects. Sensitivity nlysis is performed with respect to the voltility rte of the timber price nd the discount rte. The presented model is pplied to the study of the Koren lrch cse to identify the threshold timber price bove which it is optiml to hrvest trees. Keywords: forest vlution; crbon forestry; optiml hrvesting time; stochstic price 1. Introduction The nthropogenic crbon dioxide occurred during the lst 40 yers ccounts for bout hlf of the nthropogenic CO 2 ccumulted between 1750 nd 2010, its direct sources being energy supply, industry, trnsport nd buildings, ccording to IPCC (Intergovernmentl Pnel on Climte Chnge) 5th ssessment report. The report concludes tht the globl verge surfce temperture by 2100 will hve incresed from 3.7 C to 4.8 C compred to pre-industril levels without dditionl mitigting efforts [1]. The policies for the mitigtion of temperture rise nd the ccompnying climte chnge focus on controlling crbon emissions nd utilizing rtificil crbon sinks. Although industril sectors hve shown resistnce to emission reductions for the environment believing tht they limit profitbility nd growth, protecting the environment is essentil, even for sustinble economic growth. The emission trding scheme is mrket where the trde of crbon permits occur between the sellers who reduce crbon emission below the bseline nd the buyers who emit excessively, nd hs been designed to provide economic incentives for industril sectors to prticipte in emission reduction nd to minimize the economic inefficiency derived by controls. In the emission trding scheme, crbon offset system in which trdble permits re issued by the mount of crbon offset hs criticl role in not only chieving the emission reduction gol by using crbon sinks, but lso sustining economic growth. New Zelnd is one of the countries in which offset schemes re well-estblished. The forestry sector in New Zelnd offsets pproximtely 32% of its totl greenhouse gs emissions, proving tht non-energy sectors prticipting in emission trding schemes mke significnt contributions to reduce emissions nd to void mrket distortion [2]. A totl of 65% of Koren territory is covered with forests, with 70% of forests being privte. Efficient forest mngement will offset round 92,300 tc, which implies tht forestry will sve pproximtely 41.4 billion won [3]. In terms of forest owners, the crbon offset scheme is ttrctive Sustinbility 2018, 10, 3290; doi: /su

2 Sustinbility 2018, 10, of 13 s it provides n opportunity to mke profit while trees grow, in contrst to just mking profit from timber sles t hrvesting time. Emission permits strted to trde from Jnury 2015 in Kore. The trde of the Kore Credit Unit (KCU), which is its own offset credit, begn in April 2015, lthough the trde volume remins low. Therefore, it is criticl to provide incentives to prticipnts in the first stge of the new system to chieve its objective by nlyzing the profit structure of forest owners nd suggesting wys for efficient forest mngement. However, there hve been limited studies on crbon forest mngement in Kore, with few pproches nd methods used in the nlysis. Kyung-Tek Min (2011) empiriclly nlyzed the optiml hrvesting ge in Kore, including the benefits from crbon sequestrtion. The strong ssumption of fixed prices ws used with the Fustmnn model [4]. Sng-Min Lee et l. (2011) evluted forest vlues, combining both the economic vlue from timber hrvesting nd the environmentl vlue from crbon sequestrtion. By mximizing the socil utility of function of the mount of timber hrvested nd crbon sequestered, the optiml mount of timber hrvest cn be obtined. Although the chnge of timber volume hs been dynmiclly nlyzed, no stochstic vrible is permitted [3]. In study by Hee Sun Jng et l. (2010), the voltility of future price ws considered with the rel option pproch; crbon price following the geometric Brownin motion. However, tht ws for A/R CDM (Afforesttion/Reforesttion Clen Development Mechnism) projects in which the investment of developed countries to developing countries is dmitted s its reduction, not for Koren forest owners [5]. Models for crbon forestry reflecting relities, such s the dynmics of timber nd crbon stock chnges, the uncertinties of price movements, nd the possibilities of linked interntionl offset mrkets, need to be developed. Moreover, it is criticl tht the models re pplied to empiricl studies in Koren crbon forestry for the offset mrket. Before pplying model, the fundmentl method of forest vlution suggested by Fustmnn in 1849 should be reviewed. The net present vlue of forest is estimted by discounted future csh flows from timber sles. Bsed on the Fustmnn model, mny efforts hve been mde iming to reflect relity in model for forest vlution. Brzee, R. nd Mendelsohn, R. (1988) considered the voltility of the future price. A stochstic timber price ws incorported, nd they concluded tht the net present vlue of forest is evluted higher thn tht bsed on the Fustmnn model [6]. Englin, J. nd Ckkwy, J. M. (1993) recognized nother source of income for forest owner, considering the crbon-sequestered function of forest. They integrted the revenue from crbon credit sles, nd developed n optiml hrvesting rule [7]. Forest mngement nd the optiml cutting time of crbon forestry with stochstic price hve been nlyzed in mny studies. Chldná, Z. (2007) developed the optiml rottion rules, nd provided sensitivity nlysis for the price process types, the structure of incomes nd the discount rtes. It ws ssumed tht the timber price follows men-reverting process while the crbon price follows the geometric Brownin motion [8]. Guthrie, G., nd Kumreswrn, D. (2009) used rel option model with stochstic timber price following men-reverting process. The optiml time for cutting trees ws suggested, nd the options for replnting nd bndoning forests were exmined [9]. Tee, J. et l. (2014) further investigted the flexible vlution for crbon forestry with two stochstic prices using rel option binomil tree, llowing for joint optimiztion of the hrvest decision [10]. On top of the reviewed literture, the gols of this pper re s follows. (1) Setting up model to mximize the profit for the forest owners. The crbon credit revenue is dded to the trditionl income source of timber sles; (2) With stochstic timber price considered, the optiml time to hrvest is investigted in order to mximize the expected returns of forest mngement; (3) An empiricl study is performed with the dt of the species of lrch in Kore. 2. Methods The vlue function in forestry is the expected present vlue of forest mngement. The totl revenue is seprted into two prts; tht from the crbon credit sles, nd tht from the timber sles. The crbon revenue is clculted by integrting the future csh flow from the crbon credit sles by

3 Sustinbility 2018, 10, of 13 the mount of crbon sequestrtion of the trees up until the forest is hrvested. It is ssumed tht ll the trees re cut nd sold s timber when the forest owner determines to hrvest nd exit the forest mrket. The costs rise only t hrvesting time: The hrvesting cost nd the exit cost. The exit cost is type of penlty for relesing the stored crbon into the tmosphere. [ τ mxe p sθ V(t)e ρt dt + (X(τ) cv(τ) K)e ]. ρτ (1) 0 Eqution (1) is the objective function, which will be mximized by considering the optiml timing of hrvest. The gol is to determine the optiml hrvesting time, τ, determined endogenously. τ 0 sθ V(t)e ρt dt is the crbon revenue from the sles of the offset credits produced by the mount of nnul crbon stock chnges. The future revenues should be discounted by the discount rte, ρ. The offset credit is sold t the fixed crbon price, s. The crbon stock chnge is ssumed to be proportionl to the timber volume chnge, V(t), which decreses exponentilly; V(t) = V 0 e r1t. How much crbon is sequestered reltive to the timber growth differs by the species of the trees. Ech country tries to develop its own sequestrtion fctors for different species of trees. The sequestrtion fctor, θ, is the product of bsic wood density, the biomss expnsion fctor, the root-shot rtio nd the crbon fction. The forest owner gins the crbon sles revenue up until the trees re hrvested t the time, τ. The profit from the timber sles is (X(τ) cv(τ) K)e ρτ, which is immeditely relized t hrvesting time, τ. c is unit hrvesting cost, nd K is lump-sum penlty to exit forestry. By considering the stochstic nture of the revenue, the sles revenue, X(t), is ssumed to follow the geometric Brownin motion. dx(t) = µx(t)dt + σx(t)dz(t). (2) In Eqution (2), µ is the drift rte of the timber revenue, nd σ is the voltility rte. dt is the increment of time, nd dz(t) represents the Wiener process. The expecttion of the Wiener increment is E(dz) = 0, nd the vrince is Vr(dz) = dt. The uncertinty of the timber revenue increses s time psses becuse the vrince of dz is shown to be the time increment. The timber revenue depends on the price of timber, p(t), nd the timber volume, V(t); tht is, X(t) = p(t)v(t). It is ssumed tht the price of timber, p(t), follows the geometric Brownin motion with the drift rte of timber price, α, nd the voltility rte, σ, s shown in Eqution (3). dp(t) = αp(t)dt + σp(t)dz(t). (3) An exponentil timber growth is ssumed. In generl, the logistic function is pproprite for the growth function of trees. However, bsed on the ssumption tht hrvesting normlly occurs during the time when the trees grow exponentilly, exponentil timber growth is not regrded s n unresonble ssumption for the ske of simplicity. The timber growth function with the growth rte, r 2, is defined by V(t) = V 0 e r 2t which provides µ = α + r 2. From now on, the timber revenue, X(t), is denoted s p(t)v(t), nd we will follow the stochstic movement of the timber price, p(t). τ = in f {τ 0 p(t) = }. (4) Eqution (4) is the condition of the optiml hrvesting time. The vlue function (1) is mximized s the timber price goes beyond the threshold price tht will be numericlly obtined in the following nlysis. Note tht > p 0 holds to void the trivil problem where p 0 is the initil price of timber. The optiml time for hrvesting, τ, is the first pssge time when the stochstic timber price, p(t), is expected to hit the threshold. Let G() denote the expected present vlue of the objective function (1) tht is defined s below: τ G() = E p [ sθ V 0 e (ρ+r1)t dt + (p(τ)v 0 e r2τ cv 0 e r2τ K)e ρτ ]. 0

4 Sustinbility 2018, 10, of 13 Note tht the hrvest time τ is subject to stochstic process (3). By pplying the expecttion opertor to the prts endogenously determined, we obtin τ G() = E p [ sθ V 0 e (ρ+r1)t dt] + E p (p(τ))v 0 E p (e (ρ r2)τ ) cv 0 E p (e (ρ r2)τ ) KE p (e ρτ ). 0 Since p(τ) =, G() = E p [ τ 0 ] sθ V 0 e (ρ+r1)t dt + V 0 E p (e (ρ r2)τ ) cv 0 E p (e (ρ r2)τ ) KE p (e ρτ ). As the definite integrl is clculted, we then obtin G() = sθ V 0 ρ + r 1 [1 E p (e (ρ+r 1)t dt)] + V 0 E p (e (ρ r 2)τ ) cv 0 E p (e (ρ r 2)τ ) KE p (e ρτ ). (5) The expected discounted fctors re derived using the sme process s Chng, F. R. (2005) nd Hrrison, J. M. (1985, p. 42) [11,12]. The expected discounted fctors re s follows where β 1 = β 2 = γ = E(e (ρ+r 1)τ ) = ( p 0 )β 1, E(e (ρ r 2)τ ) = ( p 0 )β 2, E(e ρτ ) = ( p 0 )γ (α σ2 2 ) + (α σ2 2 )2 + 2(ρ + r 1 )σ 2 σ 2, (α σ2 2 ) + (α σ2 2 )2 + 2(ρ r 2 )σ 2 σ 2, (α σ2 2 ) + (α σ2 2 )2 + 2ρσ 2 σ 2. The expected discounted fctors β 1, β 2 nd γ should be greter thn one to hve intuitively correct vlues. Eqution (5) with the expected discount fctors is given s G() = sθ V 0 ρ + r 1 sθ V 0 ρ + r 1 ( p 0 )β 1 + V0 ( p 0 )β 2 cv0 ( p 0 )β 2 K( p 0 )γ. (6) The first two terms, sθ V 0 ρ+r [1 1 ( p 0 ) β 1], represent the fundmentl vlue of crbon revenue. As the threshold timber price,, increses, the crbon revenue increses. This is intuitive becuse the erning period of crbon revenue is extended. A higher threshold of timber price indictes tht the time to hrvest is delyed s we ssumed tht the timber price follows the geometric Brownin motion. The other terms in Eqution (6), V 0 ( p 0 ) β2 cv 0 ( p 0 ) β2 K( p 0 ) γ, re ssocited with timber. V 0 ( p 0 ) β2, is the timber sles revenue, nd cv 0 ( p 0 ) β2 + K( p 0 ) γ is the costs t hrvesting time. When the sles price of timber,, increses, which mens the time to hrvest is postponed, the timber revenue lso increses with the more volume of timber sold. The timber revenue in terms of future vlue grows simply, but the discounted fctors lower the timber profit s time psses. We im to determine the threshold timber price,, tht mximizes the expected current vlue of the forest, G(), nd the corresponding time, τ, s the optiml time to hrvest trees. Numericl nlysis is performed to determine the mximum point of G() solving the nonliner function, Eqution (6), which is not monotoniclly incresing or decresing.

5 Sustinbility 2018, 10, x FOR PEER REVIEW 5 of 12 Sustinbility 2018, 10, of 13 incentive for lrch forest owners to prticipte in the offset mrket s the owner my increse his profit from crbon credits sles. 3. Results The trnsction timber price of lrch from Jnury 2000 December 2013 ws sourced from yerly reports 3.1. Dt for forestry Descriptionby the Kore Forest Service (Figure 1) [15 27]. A price, ( ), follows the geometric Brownin motion nd the drift rte,, nd the voltility rte,, re estimted through log-norml trnsformtion s in Tsy (2002) [28]. Let ( ) = ( ) ( 1), nd be the time intervl. Let = 1/12 since we use monthly dt. The log difference of price, ( ), is normlly distributed with men ( 2) nd vrince. Let nd s denote the smple men nd stndrd devition respectively. Therefore, we obtin the drift rte nd the voltility rte s shown in Equtions (7) nd (8). The lrch forest is the most common mong the rtificil forests in Kore [13]. The timber of lrch is multipurpose nd of economic significnce, nd the verge crbon stock for bseline scenrio is tc/h, which is mediocre level mong mjor tree species in Kore [14]. There is n incentive for lrch forest owners to prticipte in the offset mrket s the owner my increse his profit from crbon credits sles. The trnsction timber price of lrch from Jnury 2000 December 2013 ws sourced from yerly reports for forestry by the Kore Forest Service (Figure 1) [15 27]. A price, p(t), follows the geometric Brownin motion nd the drift rte, α, nd the voltility rte, σ, re estimted through log-norml trnsformtion s in Tsy (2002) [28]. Let y(t) = lnp(t) lnp(t 1), nd be the time intervl. Let = 1/12 since we use monthly dt. The = log difference of price, y(t), is normlly distributed with men (α σ 2 /2) nd vrince σ 2. Let y + nd s2 y denote the smple men nd stndrd devition respectively. Therefore, we obtin the drift rte nd the voltility rte s shown in Equtions (7) nd (8). = α = y (8) + s2 y (7) 2 The smple men,, is , nd stndrd σ = s devition, s y, is The estimte of the (8) drift rte is given s = 0.02, nd voltility rte s =0.09. (7) Figure 1. Timber Price, Jnury 2000 December Figure 1. Timber Price, Jnury 2000 December The smple men, y, is , nd stndrd devition, s y, is The estimte of the drift The timber rte is given volume s α = dt 0.02, nd ws voltility sourced rte s from σ = the Kore Forest Service nd the Kore Forest The timber volume dt ws sourced from the Kore Forest Service nd the Kore Forest Reserch Reserch Institute (Figure 2) [29]. The originl dt uses 5-yer time intervls, showing polynomil Institute (Figure 2) [29]. The originl dt uses 5-yer time intervls, showing polynomil function; function; therefore, the dt is reproduced s nnul dt by polynomil interpoltion. therefore, the dt is reproduced s nnul dt by polynomil interpoltion.

6 The timber volume dt ws sourced from the Kore Forest Service nd the Kore Forest Reserch Institute (Figure 2) [29]. The originl dt uses 5-yer time intervls, showing polynomil function; therefore, the dt is reproduced s nnul dt by polynomil interpoltion. Sustinbility 2018, 10, of 13 Figure Figure Timber volume for for site site index index The model in Section 2 ssumed n exponentil timber growth for clcultion simplicity nd to follow existing theories of forestry. The hypotheticl increse rte of timber volume, r 2, is estimted s In order to reflect the chrcteristics of the dt, the finl results will be djusted. It is ssumed tht offset credits issued yerly re sold during the sme yer. The crbon stock chnge is bsed on the species-specific sequestrtion fctor nd the timber volume chnge, s shown in Eqution (9). Crbon stock chnge (tco2/h/yr) = V(t) D BEF (1 + R) CF (9) Tble 1 summrizes the crbon sequestrtion fctors of lrch developed by the Kore Forest Service [30]. The sequestrtion fctor, θ, for lrch is clculted to be by Eqution (9). Bsic Wood Density (D) (tdm/m 3 ) Tble 1. Crbon sequestrtion fctors for lrch. Biomss Expnsion Fctor (BEF) Root-Shot Rtio (R) Crbon Fction (CF) It is ssumed tht the nnul chnge of timber volume nd crbon stock decrese exponentilly. As shown in Figure 3, the decresing rtes of the timber volume chnge nd the crbon stock chnge show no difference becuse the crbon stock chnges re the result of the sclr multipliction of the timber volume chnge by θ. The decresing rte, r 1, is estimted s The discount rte is ssumed to be 5%. The sensitivity nlysis with regrds to the discount rte is presented below. Let the price of the offset credit, KCU (Kore Credit Unit) be constnt t 10,000 won/tco 2. Although the mount of KCU trnsctions is quite smll, it is reported by the Kore Environment Institute tht the prices re pproximtely 10,000 won/tco 2 [31]. Hrvesting cost per unit volume, c, is 62,500 won ccording to project design document (PDD) reported by the Kore Forest Service. A penlty to exit, K, is ssumed to be zero s the penlty now is zero to intrigue more foresters to join in the offset system in Kore. All the prmeters re summrized in Tble 2.

7 hown in Figure 3, the decresing rtes of the timber volume chnge nd the crbon stock chn no difference becuse the crbon stock chnges re the result of the sclr multipliction of t er volume chnge by. The decresing rte,, is estimted s Sustinbility 2018, 10, of 13 Figure 3. Timber volume chnge nd Crbon stock chnge. Figure 3. Timber volume chnge nd Crbon stock chnge. Tble 2. Prmeter estimtes. Prmeter The discount rte is ssumed to be 5%. The sensitivity nlysis with regrds to the discount r The drift rte, α 0.02 esented below. Let the price of the Theoffset voltility rte, credit, σ KCU (Kore 0.09Credit Unit) be constnt t 10,0 The negtive growth rte of V(t), r /tco2. Although the mount of KCU trnsctions The growth rte of V(t), r 2 is quite smll, it is reported by the Kor The sequestrtion fctor, θ ronment Institute tht the prices re pproximtely 10,000 won/tco2 [31]. Hrvesting cost p The discount rte, ρ 0.05 volume,, is 62,500 won ccording The offset to credit project price, s design document 10,000 won (PDD) reported by the Kor The initil timber volume, V(0) 64.5 m st Service. A penlty to exit,, is ssumed to be zero s the 3 /h The initil chnge of timber volume, V(0) 6.38 m 3 /h/yer penlty now is zero to intrig The initil price of timber, p(0) 102,000 won e foresters to join in the offset system in Kore. All the prmeters re summrized in Tble Empiricl Result Tble 2. Prmeter estimtes. Estimtes The hrvesting cost, c 62,500 won/m 3 The penlty, K 0 won Prmeter Estimtes The drift rte, 0.02 The voltility rte, 0.09 The negtive growth rte of ( ), The growth rte of ( ), Firstly, whether forest owner hs enough incentive to prticipte in the offset mrket ws investigted. Figure 4 compres the profit from both crbon nd timber sles with the profit from only timber sles. Crbon forestry yields higher profit t ny threshold timber price of ; tht is, forest owners cn ern more money if they prticipte in the offset mrket, regrdless of the timber price t which they determine to sell. The optiml threshold price will provide the mximum vlue of the forest. Figure 5 represents tht threshold price, which converges to bout 260,000 won/m 3. This implies tht the forest owner needs to decide to hrvest the trees when the timber price reches 260,000 won/m 3 in order to mximize his profits, including both crbon credits nd timber sles.

8 investigted. Figure 4 compres the profit from both crbon nd timber sles with the profit from only timber sles. Crbon forestry yields higher profit t ny threshold timber price of ; tht is, forest owners cn ern more money if they prticipte in the offset mrket, regrdless of the timber price t which Sustinbility they determine 2018, 10, 3290 to sell. 8 of 13 Figure 4. Profits of crbon forestry nd timber-only forestry. Figure 4. Profits of crbon forestry nd timber-only forestry. The optiml threshold price will provide the mximum vlue of the forest. Figure 5 represents tht threshold price, which converges to bout 260,000 won/m 3. This implies tht the forest owner needs to decide to hrvest the trees when the timber price reches 260,000 won/m 3 in order to mximize his profits, including both crbon credits nd timber sles. Figure 5. The threshold price of timber,. The corresponding first hitting time cn be obtined by: E(τ ) = { 1 α 0.5σ 2 ln ( p p 0 ) i f α > 0.5σ 2 i f α 0.5σ 2. (10) The optiml time t which the owner should determine hrvest is 45.1 yers fter the trees re plnted. Figure 5. The threshold price of timber,. The threshold price nd the corresponding optiml time to hrvest re ffected by the voltility of the price nd the discount rte. The sensitivity nlyses of the threshold price nd the optiml

9 The The threshold optiml time price t nd which the the corresponding owner should optiml determine time to hrvest hrvest is re 45.1 ffected yers fter by the the voltility trees re of plnted. the price nd the discount rte. The sensitivity nlyses of the threshold price nd the optiml hrvesting The threshold time with price respect nd to the the corresponding voltility rte optiml of the timber time to price hrvest nd the re discount ffected by rte the re voltility shown below. of the price nd the discount rte. The sensitivity nlyses of the threshold price nd the optiml Sustinbility 2018, 10, of 13 hrvesting Figure time 6 illustrtes with respect tht the to converging the voltility point rte of the threshold timber price price nd becomes the discount higher rte s the re rte shown of price below. voltility increses. The reltively high risk of business with price fluctution increses the threshold Figure of 6 the illustrtes timber price. tht the converging point of the threshold price becomes higher s the rte of price voltility increses. The reltively high risk of business with price fluctution increses the threshold of the timber price. hrvesting time with respect to the voltility rte of the timber price nd the discount rte re shown below. Figure 6 illustrtes tht the converging point of the threshold price becomes higher s the rte of price voltility increses. The reltively high risk of business with price fluctution increses the threshold of the timber price. Figure 6. Sensitivity nlysis of the threshold price (the voltility rte). Figure 6. Sensitivity nlysis of the threshold price (the voltility rte). Figure 7 shows Figure n inverse 6. Sensitivity reltionship nlysis between of the threshold the threshold price (the price voltility nd rte). the discount rte. The higher the discount Figurerte, 7 shows the n lower inverse the reltionship evluted between present thevlue threshold of crbon price ndforestry. the discount It is rte. optiml for Figure The 7 shows higher the n discount inverse rte, reltionship the lower the evluted between present the threshold vlue of crbon price forestry. nd It the is optiml discount for forest owner to determine to hrvest when the timber price is reltively low t higher discount rte. rte. The forest owner to determine to hrvest when the timber price is reltively low t higher discount rte. higher the discount rte, the lower the evluted present vlue of crbon forestry. It is optiml for forest owner to determine to hrvest when the timber price is reltively low t higher discount rte. Figure 7. Sensitivity nlysis of the threshold price (the discount rte). Figure 7. Sensitivity nlysis of the threshold price (the discount rte). The overll sensitivity nlysis of the optiml hrvesting time with respect to the voltility rte nd the discount Figure rte 7. Sensitivity providednlysis in Figure 8. of The the higher threshold threshold price price (the mens discount the time rte). for hrvest The overll sensitivity nlysis of the optiml hrvesting time with respect to the voltility rte nd the discount rte is provided in Figure 8. The higher threshold price mens the time for hrvest The overll sensitivity nlysis of the optiml hrvesting time with respect to the voltility rte nd the discount rte is provided in Figure 8. The higher threshold price mens the time for hrvest

10 needs to be postponed. The optiml hrvesting time is delyed by n increse in the price voltility nd decrese in the discount rte. Sustinbility 2018, 10, x FOR PEER REVIEW 9 of 12 Sustinbility 2018, 10, of 13 needs to be postponed. The optiml hrvesting time is delyed by n increse in the price voltility nd decrese in the discount rte. needs to be postponed. The optiml hrvesting time is delyed by n increse in the price voltility nd decrese in the discount rte. Figure 8. Sensitivity nlysis of the optiml hrvesting time (the discount rte & the voltility rte). However, this result is bsed on the ssumption of exponentil timber growth. As shown in Figure 9, gp exists between the timber volume used in the model nd the ctul timber volume Figure 8. Sensitivity nlysis of the optiml hrvesting time (the discount rte & the voltility rte). dt. Figure An djustment 8. Sensitivity process nlysis is of needed the optiml for the hrvesting tree growth time (the to be discount shown rte s & polynomil the voltility rte). function in the dt. The cubic However, polynomil this result is function bsed of thetimber ssumption growth, of exponentil timber growth. As shown in However, Figure this 9, result gp exists is between bsed the on timber the ssumption volume used in the of model exponentil nd the ctul timber growth. volume dt. As shown in An djustment process is needed for the tree growth to be shown s polynomil function in the dt. Figure 9, gp exists between the timber volume used in the model nd the ctul timber volume The cubic polynomil function ( ) of timber = growth, + + (11) dt. An djustment process is needed for the tree growth to be shown s polynomil function in the dt. The cubic polynomil function of V(t) timber = m 1 t + growth, m 2 t 2 + m 3 t 3 (11) ws estimted s in Ref. [8], nd the results re s below in Tble 3. ws estimted s in Ref. [8], nd the results re s below in Tble 3. ( ) = + + (11) ws estimted s in Ref. [8], nd the results re s below in Tble 3. Figure 9. Timber growth function. Figure 9. Timber growth function. Tble 3. The result of OLS estimtion for the polynomil timber growth. Vrible Coefficient Std. Error Figure AGE 9. Timber growth function

11 Sustinbility 2018, 10, x FOR PEER REVIEW 10 of 12 Sustinbility 2018, 10, of 13 ( ) =. (12) Tble 3. The result of OLS estimtion for the polynomil timber growth. In ddition, the threshold of the Vrible timber price, Coefficient, nd Std. the Error corresponding optiml hrvesting AGE AGE AGE time,, were clculted. Plugging these endogenously determined vribles into the vlue function (1), the mximized profit,, cn be clculted s The exponentil growth function of timber in the model is given s = 1 ( ) +[ ( ) ( ) ]. (13) + V(t) = V 0 e r 2t. (12) In ddition, the threshold of the timber price,, nd the corresponding optiml hrvesting time, τ, were clculted. Plugging these endogenously determined vribles into the vlue function (1), the mximized profit, G τ, cn be clculted s The finl objective is to find the ctul optiml hrvesting time corresponding to the timber dt showing polynomil growth. The profit clculted with the polynomil timber growth function (11),, should be equl to. G τ = sθ V 0 ρ + r 1 [1 e (ρ+r 1)τ ] + [ V(τ) cv(τ) K]e ρτ. (13) The finl objective is to find the ctul optiml hrvesting time τ corresponding to the timber dt 1 ( ) + ( ) showing polynomil + growth. The profit clculted with the polynomil timber growth function (11), G, should be equl to G τ. = ( ). (14) will be djusted to by G = numericlly sθ V 0 [1 e (ρ+r equting 1)τ ] + [ V(τ) (13) to cv(τ) (14). K]e ρτ. (14) ρ + r The gp between nd 1 ws plotted in Figure 10. At the point where the function ( ) equls zero, the new optiml hrvest time is determined. While the previous optiml hrvest time,, ws 45.1 ssuming exponentil timber growth, the djusted hrvest time,, is delyed for 7 yers to 52.1 by pplying polynomil timber growth. A totl of 14 yers should be dded s the first 14 yers of dt when the chnge of timber volume incresed hd been excluded in the nlysis. The lst djusted optiml time is τ will be djusted to τ by numericlly equting (13) to (14). The gp between G nd G τ ws plotted in Figure 10. At the point where the function (G G τ ) equls zero, the new optiml hrvest time is determined. While the previous optiml hrvest time, τ, ws 45.1 ssuming exponentil timber growth, the djusted hrvest time, τ, is delyed for 7 yers to 52.1 by pplying polynomil timber growth. A totl of 14 yers should be dded s the first 14 yers of dt when the chnge of timber volume incresed hd been excluded in the nlysis. The lst djusted optiml time is Figure 10. The gp between G nd G τ. Figure 10. The gp between nd. 4. Discussion In this study, the threshold timber price nd the optiml time for hrvest re estimted, while

12 Sustinbility 2018, 10, of Discussion In this study, the threshold timber price nd the optiml time for hrvest re estimted, while llowing stochstic timber price nd crbon revenue from offset credit sles in the cse of lrch in Kore. Crbon revenues re generted from the mount of the nnul chnge of crbon stock until forest owner decides to hrvest. Timber sles revenue rises only t the time of hrvest. Crbon forestry pprently provides more profit for forest owners thn timber-only forestry. Combining both revenue sources, the optiml hrvest time is determined endogenously to mximize the forest owner s profit, involving the time effect between the crbon revenue nd the timber revenue. The sensitivity nlyses for the voltility rte of the price nd the discount rte re lso provided. The threshold price increses s the voltility rte increses, nd the sensitivity nlysis of the threshold price with respect to the discount rte showed the opposite result; the threshold price increses with lower discount rte. In the model, n exponentil timber growth is ssumed while the timber volume dt shows polynomil growth function. The first estimte of the optiml hrvest time needs to be djusted by pplying polynomil timber growth. The djusted optiml hrvest time is delyed to yield the mximized profit. A limittion of this study is the ssumption of single rottion while mny other studies on forest vlution re bsed on n infinite rottion model. For simplicity, here, wht hppens fter hrvesting is not considered. Moreover, the crbon price is fixed in the model becuse crbon price dt hs not been sufficiently ccumulted in Kore for reserch purposes until now. As suggestion for more flexible future study on this issue, it my be interesting to llow for n infinite rottion or to consider lnd conversion option with two stochstic prices. Author Contributions: Conceptuliztion, S.Y. nd H.P.; Methodology, S.Y. nd H.P.; Writing-Originl Drft Preprtion, S.Y.; Writing-Review & Editing, Y.C. nd H.P.; Supervision, Y.C. nd H.P. Funding: This reserch received no externl funding. Conflicts of Interest: The uthors declre no conflict of interest. References 1. Edenhofer, O.; Pichs-Mdrug, R.; Sokon, Y.; Frhni, E.; Kdner, S.; Seyboth, K.; Adler, A.; Bum, I.; Brunner, S.; Eickemeier, P.; et l. IPCC, 2014: Summry for Policymkers. In Climte Chnge 2014: Mitigtion of Climte Chnge. Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmentl Pnel on Climte Chnge; Cmbridge University Press: Cmbridge, UK; New York, NY, USA, 2014; pp ISBN Kim, E.G.; Prk, S.B.; Kim, D.H. A Review on the New Zelnd Forest Emissions Trding Scheme. Koren J. For. Econ. 2008, 16, Lee, S.-M.; Kim, K.-D.; Song, S.-H. Optiml Forest Mngement Schemes with Crbon Storge Vlue Included. J. Rurl Dev. 2011, 34, Min, K.-T. An Anlysis of Forest Cutting Age with Considertion of Forest Crbon Sequestrtion. J. Rurl Dev. 2011, 34, Jng, H.S.; Prk, H.J.; Choi, Y.T. Rel Option Anlysis of A/R CDM Project Considering CER Price Uncertinty. J. Econ. Stud. 2010, 28, Brzee, R.; Mendelsohn, R. Timber Hrvesting with Fluctuting Prices. For. Sci. 1998, 34, Englin, J.; Cllwy, J.M. Globl Climte Chnge nd Optiml Forest Mngement. Nt. Resour. Model. 1993, 7, [CrossRef] 8. Chldná, Z. Determintion of Optiml Rottion Period under Stochstic Wood nd Crbon Prices. For. Policy Econ. 2007, 9, [CrossRef] 9. Guthrie, G.; Kumreswrn, D. Crbon Subsidies, Txes nd Optiml Forest Mngement. Environ. Resour. Econ. 2009, 43, [CrossRef] 10. Tee, J.; Scrp, R.; Mrsh, D.; Guthrie, G. Forest vlution under the New Zelnd Emissions Trding Scheme: A Rel Options Binomil Tree with Stochstic Crbon nd Timber Prices. Lnd Econ. 2014, 90, [CrossRef]

13 Sustinbility 2018, 10, of Chng, F.R. On the Elsticities of Hrvesting Rules. J. Econ. Dyn. Control 2005, 29, [CrossRef] 12. Hrrison, J.M. Brownin Motion nd Stochstic Flow Systems; John Wiley & Sons Inc.: New York, NY, USA, 1985; pp ISBN 10: Koo, G.-B.; Be, S.-W.; Ryu, S.-H.; Lee, C.-Y.; Kim, K.-W.; Prk, B.-W.; Hong, S.-C.; Noh, E.-W.; Hong, K.-N.; Hn, S.-U. Economic Tree Species 4 Lrix kemferi, 55; Kore Forest Reserch Institute: Seoul, Kore, 2012; ISBN Kim, Y.-H.; Jeon, E.-J.; Shin, M.-Y.; Chung, I.-B.; Lee, S.-T.; Seo, K.-W.; Pho, J.-K. A study on the Bseline Crbon Stock for Mjor Species in Kore for Conducting Crbon Offset Projects bsed on Forest Mngement Projects. J. Koren Soc. For. Sci. 2014, 103, [CrossRef] 15. Shin, S.-W. The Sttisticl Yerbook of Forestry 2001, 31; Kore Forest Service: Dejeon, Kore, 2001; p Kim, B.-I. The Sttisticl Yerbook of Forestry 2002, 32; Kore Forest Service: Dejeon, Kore, 2002; p Choi, J.-S. The Sttisticl Yerbook of Forestry 2003, 33; Kore Forest Service: Dejeon, Kore, 2003; p Cho, Y.-H. The Sttisticl Yerbook of Forestry 2004, 34; Kore Forest Service: Dejeon, Kore, 2004; p Cho, Y.-H. The Sttisticl Yerbook of Forestry 2005, 35; Kore Forest Service: Dejeon, Kore, 2005; p Suh, S.-J. The Sttisticl Yerbook of Forestry 2006, 36; Kore Forest Service: Dejeon, Kore, 2006; p Suh, S.-J. The Sttisticl Yerbook of Forestry 2007, 37; Kore Forest Service: Dejeon, Kore, 2007; p H, Y.-J. The Sttisticl Yerbook of Forestry 2008, 38; Kore Forest Service: Dejeon, Kore, 2008; p Chung, K.-S. The Sttisticl Yerbook of Forestry 2009, 39; Kore Forest Service: Dejeon, Kore, 2009; p Chung, K.-S. The Sttisticl Yerbook of Forestry 2010, 40; Kore Forest Service: Dejeon, Kore, 2010; p Lee, D.-K. The Sttisticl Yerbook of Forestry 2011, 41; Kore Forest Service: Dejeon, Kore, 2011; p Lee, D.-K. The Sttisticl Yerbook of Forestry 2012, 42; Kore Forest Service: Dejeon, Kore, 2012; p Shin, W.-S. The Sttisticl Yerbook of Forestry 2013, 43; Kore Forest Service: Dejeon, Kore, 2013; p Tsy, R.S. Anlysis of Finncil Time Series; John Wiley & Sons: New York, NY, USA, 2005; pp Goo, G.-J.; Son, Y.-M.; Lee, K.-H.; Kwon, S.-D.; Pyo, J.-K.; Lim, S.-S.; Yoon, H.-J. Volume Biomss nd Stnd Yield Tble; Kore Forest Service & Kore Forest Reserch Institute: Seoul, Kore, 2012; p ISBN Kore Forest Service. Avilble online: /fcm/ui_fcs_ html&mn=kfs_02_10_12_20_30&orgid=fcm (ccessed on 13 September 2018). 31. Prk, K.-G. KEI Focus: The Stte nd the Issue of Emission Trding Scheme, 3; Kore Environment Institute: Sejong, Kore, by the uthors. Licensee MDPI, Bsel, Switzerlnd. This rticle is n open ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution (CC BY) license (