Carbon Dynamics and Optimal Forest Rotation

: Carbon sequestration in forests is being considered as a mechanism to slow or reverse the trend of increasing concentrations of carbon dioxide in the atmosphere. This paper presents the results from a dynamic programming model used to determine the optimal harvest decision for a forest stand that provides both timber harvest volume and carbon sequestration services in the Canadian boreal forest. The primary contribution of this study is that the model incorporates detailed information on carbon stocks in dead organic matter pool (DOM), which have been ignored in stand level economic models. It also considers varying levels of initial DOM stocks. The varying DOM stock has interesting implications because it allows one to establish an initial DOM stock for which a landowner will voluntarily participate in the type of carbon market considered in this study. The results of the study indicate that while optimal harvest age seems to be relatively insensitive to carbon stocks in DOM pool, initial carbon stock levels signiﬁcantly aﬀect economic returns to carbon management.


Introduction
In response to global concern about climate change, policy makers and scientists are searching for ways to slow or reverse the trend of increasing concentrations of greenhouse gases, especially carbon dioxide (CO 2 ), in the atmosphere. Forests are viewed as potential carbon sinks. As trees grow, photosynthesis converts CO 2 into cellulose and other plant material, temporarily removing it from the atmosphere. In addition, a substantial amount of carbon is stored in forests as dead organic matter (DOM) in standing snags, on the forest floor, and in the soil until the process of decomposition releases it back to the atmosphere.
The Intergovernmental Panel on Climate Change (IPCC) provides guidelines for the calculation and reporting of changes in stocks of forest carbon (IPCC, 2006) as it relates to national greenhouse gas inventories. The IPCC identifies three tiers for reporting changes in stocks of forest carbon. These tiers reflect the relative importance of forest carbon stocks to greenhouse gas inventories and the sophistication of the data collection and monitoring infrastructure of countries. Canada has elected to use tier 3 methodologies (with the most detailed reporting requirements) for reporting changes to carbon stocks on managed forest lands. The IPCC specifies five carbon pools that must be accounted for: above-ground biomass, below-ground biomass, dead wood, litter, and soil carbon. The Canadian Forest Service developed the Carbon Budget Model of the Canadian Forest Sector (CBM-CFS3) to track and report changes in forest carbon stocks (Kull et al., 2007). CBM-CFS3 is a detailed model that recognizes more than 20 different carbon pools within a forest stand and tracks the transfer of carbon between these pools and the atmosphere (Fig.1).
Carbon sequestration services have value and markets are developing to allow for trade in these services. For example, the Chicago Cli- Very fast, fast, medium, and slow refer to relative decomposition rates for pools.
Curved arrows represent transfers of carbon to the atmosphere, and straight arrows represent transfers from one pools to another. SW is softwood, HW is hardwood, AG is above ground, and BG is below ground. Illustration courtesy of the Canadian Forest Service, reproduced with permission from (Kull et al., 2007, Fig.1-1). mate Exchange (CCX) has developed contracts for voluntary carbon offsets generated through forest management activities (CCX, 2009).
However, a component that has not been investigated in detail in the literature on forest carbon sequestration is the role of DOM. As the relationships between DOM, carbon stocks and forest harvesting are complex, detailed investigation of these issues is required to be assured that markets for forest carbon are accurately capturing sequestration dynamics. The purpose of this study is to explore how markets for forest carbon sequestration services could affect optimal forest management decisions when DOM is taken into account.
The classic problem in forest economics is the determination of the harvest age for an even-aged forest stand which maximizes the net present value of an infinite series of timber regeneration, growth, and harvest cycles. Faustmann (1849) is usually attributed with the first correct solution to this problem when only timber values are considered. Samuelson (1976) provides a more formal mathematical specification of the problem. Hartman (1976) extends the model to include values associated with standing trees (e.g. wildlife habitat) as well as the extractive value of timber harvest.
The Hartman model is used by van Kooten et al. (1995) in an early exploration of the effect of carbon prices on optimal forest harvest age in western Canada. In their analysis, the amount of carbon stored in the forest stand is proportional to volume of merchantable timber on the site at a particular stand age. The forest owner is paid for the accumulation of carbon in biomass associated with growth, and pays for carbon released to the atmosphere at harvest. Some of the harvested timber is assumed to be permanently stored in structures and landfills. There is no recognition of DOM or soil carbon in the van Kooten analysis.
However, DOM and soil carbon can represent a substantial proportion of the total carbon stored on forested site. Also, management decisions such a harvest age can have a substantial effect on soil carbon stocks (Aber et al., 1978, Kaipainen et al., 2004. Covington (1981) found that forest floor mass declines sharply following harvest, with about 50% of forest floor organic matter lost in the first 20 years. DOM may increase immediately following harvest as a result of slash and other debris left on site (Black and Harden, 1995). Gutrich and Howarth (2007) develop a simulation model of the economics of timber and carbon management for five different forest types in New Hampshire, USA. Their model includes representation of car-bon stored in live biomass, dead and downed wood, soil carbon, and wood products (hardwood pulp, hardwood saw, softwood pulp, and softwood saw products). Annual transfers of carbon between pools are modelled using straightforward functions. For each timber type, an initial stock of carbon in the dead and downed wood pool is assumed, and the NPV maximizing model is solved using a grid search procedure given an initial stock of carbon in the non-biomass pools.
At the forest level, McCarney (2007)  The contribution of the present study is an extension the work of van Kooten et al. (1995) incorporating DOM or soil carbon pool. While Gutrich and Howarth (2007) did include DOM and soil carbon pools in their model, they assumed a fixed initial stock of DOM. This paper extends the Gutrich and Howarth (2007) approach by solving simultaneously for a range of initial DOM stocks. As it is demonstrated later in this paper, the size of the initial DOM stock controlled by a forest owner significantly affects the incentives associated with carbon management and the attractiveness of carbon markets to forest landowners.
In this paper, a dynamic programming model is developed to determine the optimal harvest decision for a forest stand described by two state variables: stand age and carbon stocks in DOM. This model is used to: 1. examine the sensitivity of optimal harvest age to stocks of carbon in DOM, carbon prices, and lumber prices, 2. examine the sensitivity of the net present value of forested land to stand age, stocks of carbon in DOM, and carbon prices, 3. examine projected trajectories of carbon stocks in DOM given optimal harvest rules for different carbon prices, and 4. examine the impact of ignoring carbon stocks in DOM on the optimal harvest decision.

The Model
The basic assumption of the model is that the decision maker (a forest landowner) is participating in a carbon market where the landowner is paid for carbon sequestered by the forest and pays when carbon is released from the forest. The landowner is assumed to manage the forest jointly for timber production and carbon sequestration in a manner that earns maximum discounted financial return. The forest is managed using an even-aged silvicultural system. Each rotation begins with the establishment of a stand on bare forest land and ends with a clearcut harvest after a number of years of growth. The beginning of a new rotation coincides with the end of the previous rotation. The cycle of establishment, growth, and harvest is assumed to repeat ad infinitum. A tabular representation of the merchantable timber yield table from TIPSY is approximated using a Chapman-Richards growth function: in which V (t) represents the merchantable timber volume in m 3 /ha at age t and v 1 , v 2 and v 3 are parameters, which were set at 500, 0.027 and 4.0 respectively. The Chapman-Richards approximation is a good representation of the yield table generated by TIPSY (Fig.2). This is partly because of the rising Canadian dollar and cheap lumber prices. In the short run it is possible that firms might operate even with negative stand value. Stands are assumed to be reestablished immediately following harvest at a cost of 1,250 CAD/ha. The costs reported on a CAD/ha basis are assumed to be closely related to the area harvested; the costs reported as CAD/m 3 are assumed to be more closely related to volume harvested.

Accounting for carbon sequestration
The decision problem is represented as a dynamic program, and state variables are used to describe the system at each stage of the decision problem. It is theoretically possible to develop a dynamic program with state variables representing carbon stock in each of the 20+ carbon pools represented in CBM-CFS3, but program solution becomes impractical due to Bellman's "curse of dimensionality" (Bellman, 1961).
A simplified representation of the carbon pool structure of CBM-CFS3 is created using two carbon pools: a biomass pool representing carbon stored above and below ground in living trees, and a dead organic matter pool representing all other carbon stored in standing dead trees, on the forest floor, and in the soil. The label "dead organic matter" is used even though it is recognized that some of the carbon in this DOM pool is contained in living organisms.
Transfer of carbon from the biomass to the DOM pool is described by a litterfall rate expressed as a proportion of biomass, and transfer of carbon from the DOM pool to the atmosphere is described as a decay rate expressed as a proportion of DOM stocks. Under the approach used in this study, total ecosystem carbon sequestration, T EC t measured in tonnes of carbon per hectare (tC/ha) at time t, is simply the sum of the carbon in the living biomass and the DOM pools.
where B t represents total carbon sequestered in the living biomass; and DM t measures the total carbon sequestered in the DOM pool.
The yield curve produced by TIPSY was used as input to CBM-CFS3 to develop a projection of the stocks stored in the various carbon pools for the lodgepole pine stand. as a result of decay processes (Kurz et al., 1992). As the stand ages, TEC stocks increase with increasing biomass, and the decline in DOM stocks slows and reverses as carbon is added to the DOM pool in the form of litterfall, dead branches and natural tree mortality. The dynamics of the DOM pool are represented by [3].
where DM t is the amount of DOM in period t and DM t−1 is the amount of DOM in the period t − 1. B t−1 is the carbon in living biomass in period t − 1. The decay and litterfall rates were estimated using the method of least squares to find the curve which best fits the DOM curve shown in   (Kurz et al., 1992). However, this simplified representation is deemed adequate for the purposes of the current exposition.
When timber harvest occurs, the merchantable timber volume is removed from the site and processed into lumber and wood chips. The roots, stumps, tops, branches and leaves are assumed to die at the time of harvest and become part of the DOM pool. The difference between the pre-harvest stock of carbon in living biomass (B t ) and the carbon that is removed from the forest in the form of harvested wood (M C t ) is transferred to the DOM pool. Carbon removed from the forest is calculated as: where γ = 0.2 is the estimated average carbon content of wood (tC/m 3 ), and H t is the volume removed at harvest (m 3 ). The carbon content of wood varies with tree species, although it is generally in the range of 200 kg m −3 (Jessome, 1977). In the absence of timber harvest, the amount of debris transferred D t , assumes a value of zero.
The amount of debris transferred to the DOM pool can generally be expressed as: Therefore, if the amount of debris transferred after harvest is taken into account, the amount of carbon in the DOM pool at any time can be recalibrated using [3]:

Valuation of carbon
The carbon market assumed for this article pays landowners for net sequestration of CO 2 and requires payment for net release of CO 2 in the previous year. The price received per tonne of sequestered CO 2 is the same as the price paid per tonne of released CO 2 . Net carbon sequestration is calculated as the change in total ecosystem carbon stocks: In this study, a broad range of prices for CO 2 is used in sensitivity analyses. It is assumed the price of CO 2 ( P CO2 ) to be 0, 1, 2, 5, 10, 20 or 50 CAD/t of CO 2 sequestered or released.
because the molecular weight of CO 2 is approximately 3.67 times the atomic weight of carbon.

Dynamic Programming Model
The model developed here is a discrete backwards recursion dynamic programming model. The stages represent time, in one year time steps. The forest stand is described by a combination of two state variables, the age of the stand (years) and carbon stocks in the DOM pool (tC/ha). There are 251 discrete age classes, ranging from ages 0 to 250 years and 501 discrete DOM classes ranging from 0 to 500 tC/ha.
The 250 year and 500 tC/ha classes are trapping classes that represent age 250 years and older and DOM carbon stocks of 499.5 tC/ha and greater. Timber harvest volume and carbon stored in the biomass pool are calculated as a function of stand age.
At each point in time, a decision is made by the landowner whether to clearcut the stand or let the stand to grow for another year. Clearcutting yields immediate timber revenue. Both the clearcut and the leave decisions will result in a change in TEC and the appropriate carbon credit or debit. If harvesting does occur (i.e. decision, h = 1) in stage t, it is assumed that replanting occurs immediately and the stand age is reset to 1 in stage t + 1. If harvesting does not occur (i.e. decision, h = 0) in stage t, the stand age is incremented by one year in stage t + 1.
The stage return or periodic payoff (N t ) is calculated as shown in equation [9]. The payoff is calculated for the midpoints of each DOM class (d i ) and stand age (a j ) and for each of the possible harvest decisions (h). If a stand is not harvested (h = 0), the periodic payoff would be the carbon credit or debit occurring over a growth period. On the other hand if the stand is harvested (h = 1), a net harvest revenue is received from the sale of timber and either a debit is charged for net carbon loss or a credit is received for net carbon sequestered.
where a j = age of a stand in state j, and j = 0, 1, . . . , 250 years (with harvest, the age of the stand is set to 1 for the subsequent year, so the Because the objective function is to maximize net present value of timber and carbon, the discount rate becomes an important parameter. The discount factor δ shown in equation [10] represents the relative value of a dollar received one year from now (given an annual discount rate of i) to a dollar today. The discount rate used for the analysis is 5% per annum. This is the rate used in van Kooten et al. (1995) and is intended to reflect a market rate of time preference.
The return for the last stage in the problem is initialized to zero as defined in equation [11] : This assumption is justified on the basis that T is large (500 years) and the discounted value of R T for reasonable discount rates for this problem is near zero (e.g. the present value of 1 CAD received 500 years in the future is 2.543 x 10 −11 CAD) for a 5% discount rate.
The recursive objective function for this problem is given in equation [12]. The recursive function also includes the stage return function given in equation [9] and the appropriately discounted returns for each of the possible harvest decisions: . . , 2, 1. [12] The recursive objective function defines the objective function for the stages T − 1 down to 1 for each possible combination of state variables.
It calculates a return for each of the harvest decisions and selects the harvest decision that results in the maximum return as the optimal choice for the state combination in that stage.
Economic analyses for timber production management only were also conducted to produce a baseline for this study. The results from the different sets of runs are presented in the next section.

3.Results and Discussion
In order to characterize the behaviour of a landowner, a number of analyses have been undertaken. The main objective has been to analyze the sensitivity of optimal harvest decision to DOM stocks. Using a base lumber price of 375 CAD/MBF and a discount rate of 5%, attention was focused on four issues. First, the study tried to understand how alternative carbon prices will impact the optimal decision to harvest at different levels of DOM stocks. Second, the sensitivity of net present value to initial states was investigated. Third, a sensitivity analysis based on different lumber prices was performed. Fourth, and finally, the effect of ignoring DOM in the model was investigated to allow for direct comparison with the work by van Kooten et al. (1995). The results presented in this section were calculated using an implementation of the dynamic programming model programmed in MATLAB (Pratap, 2006).
The main results of this study are presented in Figures 6-8. First the results of the case where carbon sequestration services have no value to the landowner (i.e. P CO2 = 0) is discussed: the case of the Faustmann rotation. Figure 6 shows that the optimal harvest age for the base case scenario is 82 years. As carbon has no value, it is unsurprising that this harvest age is unrelated to the amount of carbon stored in the DOM pool.
In figure 7a, the value of land, standing timber and carbon sequestration services (LTCV, hereafter) is displayed for combinations of initial stand age and carbon stocks in the DOM pool. As expected, LTCV is unrelated to carbon stock when P CO2 is zero, and increases with increasing age and, therefore, the volume of standing timber. Note that LTCV is negative below a stand age of 55 years. This is consistent with other results (Adamowicz et al., 2003, Rodrigues, 1998 showing negative returns on timber growing investments in the boreal forest of Canada. Figure 8a shows the development of the stand in state-space assuming the optimal decision rule is followed. Given the starting stock in the DOM pool of 370 tC/ha, an overall decline to a dynamic equilibrium is observed with about 275 tC/ha at stand establishment and 200 tC/ha at the optimal harvest age of 82 years. This equilibrium is essentially reached after 3 rotations. Model runs were conducted using a wide range of carbon prices (P CO2 = 0, 1, 2, 5, 10, 20, 40 CAD/tCO 2 ). In the interest of saving space,  results for a subset of these prices are presented. In the next few paragraphs, Figures 6-8 examine the effects of carbon prices on the optimal harvest decision. From Figure 6 it is apparent that higher carbon prices are associated with older optimal harvest ages. In the base case scenario, the optimal harvest decision for carbon prices greater than 40 CAD/tCO 2 is to never harvest. The optimal policy is sensitive to DOM stocks at the lower levels. This happens because the amount of CO 2 released to the atmosphere through decomposition is lower with lower   LTCVs because a fixed decay rate leads to more CO 2 being released from a large stock of DOM than a small stock. As carbon prices increase, there is an increase in LTCV for stands with a low DOM carbon stock, and a decrease in LTCV for stands with a high DOM carbon stock. For the examples presented here, the boundary between low and high DOM carbon stock stands is roughly between 100 and 250 tC/ha, well below the boreal forest average of 340 tC/ha. Figure 8 shows the trajectory of the stand in state-space given implementation of the optimal decision rule determined for alternative P CO2 .
In general, the optimal harvest age increases with increasing P CO2 and reaches a point where the optimal harvest decision is to never harvest.
At equilibrium, the DOM carbon stocks are higher with higher P CO2 , but the difference is not substantial until the price is high enough for a never harvest optimum.
For completeness sake, sensitivity analysis was conducted using alternative lumber prices. Figure 6 presents the optimal harvest age by DOM carbon stocks and P CO2 for lumber prices of 375, and 500 CAD/MBF. As the lumber price increases, the optimal harvest age shortens. This is consistent with previous optimal rotation analysis, and is unsurprising. At the low lumber price (i.e., 250 CAD/MBFnot shown), the never harvest decision becomes optimal for P CO2 of 5 CAD/tCO 2 or greater; for the high lumber price, the never harvest decision is optimal for P CO2 of 75 CAD/tCO 2 or greater.
A major difference between this study and those of van Kooten et al.
(1995), Spring et al. (2005a,b), Chladná (2007), and Yoshimoto and Marušák (2007) is that this study considers carbon stored in biomass and DOM pools, whereas they ignore biomass pool. In order to evaluate the effect of ignoring the DOM pool, a series of model runs were conducted with a modified version of this model where the carbon market considered only biomass carbon. The results are summarized in Table   1.
In general, the optimal harvest age is older when the DOM pool is ignored. The older harvest age is probably related to the fact that the amount of carbon released to the atmosphere through decay is greater when the stock of carbon in DOM is considered. The optimizing model will choose a younger harvest age in order to minimize the financial Table 1. Summary of harvest ages for "with" or "without" DOM pool. penalties associated with carbon loss through decay.

Conclusions
This study presents the formulation of, and results from, a dynamic programming model used to determine the optimal harvest decision for a forest stand used to provide both timber harvest volume and carbon sequestration services. The forest stand is described using two state variables: stand age and the stocks of carbon stored in the DOM pool.
This study is unique in that it examines the impact of varying DOM on the optimal harvest age. This study provides a basic framework for assessing the economic implications of alternative methods of accounting for carbon stocks in DOM The model is used to examine optimal harvest decisions for a lodgepole pine stand in the boreal forest of western Canada. The following main conclusions can be drawn from the study: 1. Optimal harvest age increases with increasing carbon price, 2. The optimal decision is sensitive to the stocks of carbon in the DOM pool, especially when carbon prices are high and initial DOM stocks are low.
3. The optimal harvest age is younger when the dynamics of carbon in DOM are considered than when they are ignored.
4. In a market where a forest landowner is required to pay for CO 2 emissions from the forest, the carbon market may well reduce This article presented the results of an optimal harvesting model for a forest stand where the landowner is paid for net increases in total ecosystem carbon in the stand, and pays for net decreases, on an annual basis. By approximating a detailed carbon budget simulation model using two carbon pools, a dynamic programming model of the system is developed to capture the important elements of the system for an economic analysis. A variants of this model will be developed in the near future to explore alternative forms of carbon markets, including one which accounts for carbon pools in forest products.