The functional form for the yield function is
Va,ti(m a,ti) = h*[exp(δi-(πi)/a)]
(3)
Per equation h = φi(1 + ma,ti)τ, h is the stocking density, which can be adjusted depending on the intensity of management, ma = 1,ti. We restrict stocking elasticity, τ, to be positive and less than 1. The τi affects the elasticity of management inputs in forestry to account for technology change. Initial stocking is denoted by φi. Increase in ma = 1,ti will increase h, e.g., dh/dZ > 0, but the increase diminishes as ma = 1,ti rises, e.g., d2h/dZ2 < 0. The model chooses management intensity by optimally choosing ma = 1,ti. Increases in management intensity will increase yield and shift the entire yield function upward. Forests are assumed to grow according to Va,ti(Za = 1,ti), where δ and π are species-dependent growth parameters (fig. S3 shows a representative yield function assuming h = 1.32, δ = 5.2, and π = 30). C(Qttot) is the cost function for harvesting and transporting logs to the center (mills or power plants) from each of timber type.
Xa,ti = Xa-1,t-1i-Ha-1,t-1i+Ga = 0,t-1i+Na = 0,t-1i
(4)
The initial stocks of land Xti are given, and all choice variables are constrained to be greater than or equal to zero, and the area of timber harvested Ha,ti does not exceed the total timber area. Gti is the area of timber regenerated land planted, and Nti is the new forest planted. CGi(·) is the cost function for planting land in temperate and previously inaccessible forests, while CNi(·) is the cost function for planting forests in subtropical plantation regions.
GTM takes into account the competition of forestland with crops and livestock using a rental supply function for land (2). In Eq. 1, Rti(·) is the rental cost function for the opportunity costs of holding timberland Xa,ti. For example, if timber prices rise relative to agricultural land prices, the model predicts that timber owners will rent suitable farmland for at least a rotation. Similarly, if timber prices fall relatively to agricultural land prices, suitable forest land will be converted back to farmland upon harvest. In addition, the model accounts for the aggregate global effects of moving land between forests and agriculture by shifting all of the land supply functions for individual forest types as a function of the aggregate global area of forestland. As more land globally moves from agriculture to forests, all rental functions shift inward, making it more costly to convert any land from agriculture to forestry. This captures the effect that having less land in agriculture would have on land prices. We have assumed that the elasticity of land supply is 0.25, such that a 0.25% reduction in the global area of agricultural land would cause all land rents to rise by 1%. Note also that the rental supply function is restricted to agricultural land that is naturally suitable for forests. It presumes that the least productive crop and pasture land will be converted first and that rental rates increase as more land is converted and sendogenous.
The model is also developed to account for the global forest carbon stocks and flows following a method first presented by Sohngen and Sedjo (50) and updated by Daigneault et al. (18). Carbon is tracked in four pools: aboveground carbon, soil carbon, forest product carbon, and slash.
Aboveground carbon Ca,ti accounts for the carbon in all components of the living tree, including roots, as well as carbon in the forest understory and the forest floor, but does not include dead organic matter in slash, which is contained in a separate pool. For this analysis, we assume that carbon is proportional to total biomass, such that carbon in any forest of any age class is given as
Ca,ti = σi Va,ti(mt0i)
(5)
where σi is a species-dependent coefficient that converts biomass to carbon. Given this, the total forest carbon pool TFCPti for each timber type is calculated as
TFCP ti = sum over a of Ca,tiXa,ti
(6)
Carbon in harvested forest products HCti is estimated by tracking forest products over time as follows
HC ti = (1-τt)κi sum over a of (βiVa,tiHa,ti)
(7)
where κi is the proportion of harvested timber volume that is carbon stored permanently, and it is estimated to be 0.30 (51), βi is a parameter that converts forest products into carbon (regional and forest type based), while τt is the portion of wood used in the energy sector, and it is endogenously selected by the model; that is, HCti accounts only for carbon stored in wood products, not woody biomass used for energy production. Carbon stored in woody biomass used for bioenergy production is calculated as follows
BIOCti = τt sum over a of βi(Va,tiHa,ti)
(8)
Soil carbon SOLCti is measured as the stock of carbon in forest soils of type i in time t. The value of K, the steady-state level of carbon in forest soils, is unique to each region and timber type. The parameter μi is the growth rate for soil carbon. In this analysis, we capture the marginal change in carbon value associated with management or land use changes. When land use change occurs, we track net carbon gains or losses over time as follows
SOLC t+1i = SOLC ti+ SOLC ti(upmui)[((K - SOLC ti))/(SOLC ti)]
(9)
Last, we measure slash carbon AS ti as the carbon leftover on site after a timber harvest
AS ti = sum over a of (upomegaaiVa,tiHa,ti-upkappaiVa,tiHa,ti)
(10)
Over time, the stock of slash SP ti builds up through annual additions and decomposes as follows
SP t+1i = AS ti+(1-varthetai SP ti)
(11)
Decomposition rates varthetai differ, depending on whether the forest lies in the tropics, temperate, or boreal zone.
Last, in Eq. 1, the term CCt,RCP,policy represents the carbon payments/penalties for forest owners according to the policy implemented. For this study, GTM has been enhanced to capture public policy efforts that either penalize bioenergy demand or value climate mitigation benefits of forests sequestration. GTM assumes that the incentives in the timber product, woody biomass, and carbon sequestration system can be implemented efficiently. That is, GTM portrays an ideal
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The stock of land in each forest type adjusts over time according to
understory and the forest floor, but does not include dead organic matter in slash, which is contained in a separate pool. For this analysis, we assume that carbon is proportional to total biomass, such that carbon in any forest of any age class is given as