CARBON CREDITS COMPETE POORLY WITH AGRICULTURAL COMMODITIES IN AN OPTIMIZED MODEL OF LAND USE IN NORTHERN CALIFORNIA

Nascent US carbon markets reward farmers for reforesting agricultural land, with consequent ecological co-benefits. We use a dynamic optimization model to determine the likelihood of an orchard farmer in northern California converting to forest under 90 plausible future scenarios. We find reforestation to be a highly unlikely outcome, occurring only 4.0% of the time under current economic, biophysical, and policy conditions, and only 18.5% of the time under a set of assumptions that make carbon offset production more economically viable. Conversion to “carbon farming” was more sensitive to changes in orchard production costs and yields than to carbon offset policy changes. In the absence of other changes, the price of a carbon offset would have to increase nearly a hundredfold to make reforestation compete economically with orchard agriculture. Our results partly explain low participation in the reforestation sector of US carbon markets. We conclude that farmers will not be interested in forest conversion unless their land has limited agricultural potential or they are motivated by social, rather than economic, rewards.


Almond orchard
In California's Central Valley, a typical almond orchard covers 40 acres. We assume the rest of the farm, 60 acres, is used to grow continuous wheat. An almond orchard does not produce any marketable fruit until its third year and does not typically reach its maximum yield levels until its 6 th year. A typical almond orchard has a lifetime (rotation) of 25 years. If the farm remains in an almond orchard for 25 years then at the end of the 25 th year all the almond trees are ripped out of the ground. Then either new almond trees are planted or the farmer converts to a new land use.
The first year of an orchard is mainly site prep and tree planting (there is some tree planting in the second year as well). Pruning begins in the third year. Bee hives are rented every year beginning in the third year. A custom harvester is rented by the farmer each year beginning in the fourth year (third year harvest is done manually with poles by contracted labor). Every year fertilizer, various pesticides (herbicides, pesticides, and rodenticides), and irrigation water are applied to the orchard. Flood irrigation is used on almond orchards.
Annual per acre operating costs in an almond orchard is given by, where Plant At(s) is the per acre orchard planting and site prep cost in year t given the orchard is in its s th year (Plant At(s) = 0 for s ≥ 3), Pest At(s) is the per acre orchard pesticide cost in year t given the orchard is in its s th year, Fert At(s) is the per acre orchard fertilizer cost in the year t given the orchard is in its s th year, IrrWater At(s) is the per acre irrigation water cost in year t given the orchard is in its s th year, 8 Prune At(s) is the per acre orchard pruning cost in year t given the orchard is in its s th year (Prune At(s) = 0 for s ≤ 2), Pollin At(s) is the per acre cost of renting beehives in year t given the orchard is in its s th year (Pollin At(s) = 0 for s ≤ 2), Sanitation At(s) is the per acre cost of removing mummy nuts from trees in year t given the orchard stand is in its s th year (Sanitation At(s) = 0 for s ≤ 2), VehCost At(s) is the per acre cost of operating a Truck and ATV in year t given the orchard is in its s th year, Interest At(s) is the per acre interest payment on operating capital cost in year t given the orchard is in its s th year, 9 Harvest At(s) is the per acre orchard harvest cost in year t given the orchard is in its s th year (Harvest At(s) = 0 for s ≤ 2), and Carbon At(s) is the per acre tax on net carbon emissions from an almond orchard in year t given the orchard is in its s th year. The value of s cannot exceed 25. Any paid labor and equipment repair costs are included in these variables.
Annual net operating revenue on the almond orchard in year t in its s th year of rotation is equal to, where ‫‬ ௧ is the market price for a ton of almonds in year t, ܻ ௧(௦) is the tons of almonds yielded per ace in year t given the orchard is s years old, and ‫ܥ‬ ௧(௦) (equation 1) is the orchard's per acre operating cost in year t given the orchard is s years old.
The one capital cost that is unique to almond farming (and therefore not ignored) is the flood irrigation system. While an almond orchard can be abandoned after 5, 10, 15, or 20 years of operation we assume the loan taken out to purchase the flood irrigation equipment at the time of orchard establishment is sunk (and the equipment cannot be sold to another farmer to partially recover costs). We assume the flood irrigation system covers 100 acres 10 , has a 25 year lifespan, costs $101,500, and can be paid with a 25-year loan with an interest rate of 4.75.
According to the Excel function PMT(0.0475,25,101500) this means an annual payment of $7,022 / year (or $70.22 / acre / year). Because we assume no re-sale of irrigation systems the equipment is always available to the farmer for 25 years after purchase. Therefore, for example, if the farmer establishes an almond orchard in year t = 0, abandons it after 10 years, and then begins an almond orchard again at t = 19 she still has 5 years of flood irrigation equipment before she has to purchase and install a fresh irrigation system.
The removal of almond trees from the farm at the end of the 25 year rotation or earlier if the farmer abandons almond farming for another use is also attributed to the almond operation in our model. According to the almond enterprise budget, tree removal is $478 per acre.
Annual net revenue on the almond orchard (ignoring the common costs noted in the introduction) in year t given the orchard is in its s th year of operation is equal to, where ܴ ௧ is the almond tree removal cost in year t (ܴ ௧ = 0 in all years except an almond orchard's last) and ‫ܭ‬ ௧ is the year t irrigation equipment cost on an almond orchard and the remaining wheat acres. ‫ܭ‬ ௧ needs to be positive every year an almond orchard is on the land. Further, in some years ܻ ௧(௦) and ‫ܥ‬ ௧ሺ௦ሻ could equal 0 but ‫ܭ‬ ௧(௦) > 0 if the farm is no longer in almond production but the almond irrigation equipment loan is still being paid off.

Walnut orchard
In California, a typical walnut orchard covers 60 acres. We assume the rest of the farm, 40 acres, is used to grow continuous wheat. A walnut orchard does not produce any fruit until its fourth year and does not typically reach its highest annual yield levels until its 8 th year. A typical walnut orchard has a lifetime (rotation) of 25 years. If the farm remains in a walnut orchard for 25 years then at the end of the 25 th year all the walnut trees are ripped out of the ground. Then either new walnut trees are planted or the farmer converts to a new land use.
The first year is mainly site prep and tree planting (there is some tree planting in the second year as well). Pruning begins in the third year. The farmer rents a custom harvester each year beginning in the fourth year. The farmer also pays for hulling and dehydrating of the harvested nuts at an offsite facility. Every year fertilizer, various pesticides (herbicides, pesticides, and rodenticides), and irrigation water is applied to the orchard. Sprinkler irrigation is used on walnut orchards.
Annual per acre operating costs in the walnut orchard is given by, where Plant Wt(s) is the per acre orchard planting and site prep cost in the s th year of the orchard (Plant Wt(s) = 0 for s ≥ 3), Pest Wt(s) is the per acre orchard pesticide cost in the s th year of the orchard, Fert Wt(s) is the per acre orchard fertilizer cost in the s th year of the orchard, IrrWater Wt(s) is the per acre irrigation water cost in the s th year of the orchard, Prune Wt(s) is the per acre orchard pruning cost in the s th year of the orchard, VehCost Wt(s) is the per acre cost of operating a Truck and ATV in year t given the orchard is in its s th year, Interest Wt(s) is the per acre interest payment on operating capital cost in year t given the orchard is in its s th year, 11 Harvest Wt(s) is the per acre orchard harvest cost in the s th year of the orchard (Harvest Wt(s) = 0 for s ≤ 3) 12 , Carbon Wt(s) is the per acre tax on net carbon emissions from a walnut orchard in year t given the orchard is in its s th year. s cannot exceed 25. Any paid labor and equipment repair costs are included in these variables.
Annual net operating revenue on the walnut orchard in year t given that it is in its s th year of operation is equal to, where ‫‬ ௐ௧ is the market price for a pound of (in-shell and dry) walnuts in year t, ܻ ௐ௧(௦) is the tons of walnuts (in shell and dried) yielded per ace in year t given the orchard is s years old, and ‫ܥ‬ ௐ௧(௦) (equation 4) is the orchard's per acre operating cost in year t given the orchard is s years old.
The one capital cost that is unique to walnut farming (and therefore not ignored) is the pump/well and sprinkler irrigation system. While a walnut orchard can be abandoned after 5, 10, 15, or 20 years of operation we assume the loan taken out to build the pump/well and purchase the sprinkler irrigation equipment at the time of orchard establishment is sunk (and the equipment cannot be sold to another farmer to partially recover costs). We assume the irrigation system covers 100 acres 13 , has a 25 year lifespan, costs $84,000 x (100/60), and can be paid with a 25-year loan with an interest rate of 4.75. Further, we assume the pump/well has a 25 year lifespan and has a price of $70,000 x (100/60), and can be paid with a 25-year loan with an interest rate of 4.75. According to the Excel functions PMT(0.0475,25,84000 x (100/60)) and PMT(0.0475,25,70000 x (100/60)) this means annual payments of $9,686 and $8,072. Because we assume no re-sale of irrigation systems, the walnut pump/well and irrigation equipment is always available to the farmer for 25 years after purchase. Therefore, for example, if the farmer establishes an walnut orchard in year t = 0, abandons it after 10 years, and then begins a walnut orchard again at t = 19 she still has 5 years of a pump/well and sprinkler irrigation equipment before she has to purchase and install a fresh irrigation system.
The removal of walnut trees from the farm at the end of the 25 year rotation or earlier if the farmer abandons walnut farming for another use is also attributed to the walnut operation in our model. According to the walnut enterprise budget, tree removal is $150 per acre.
Annual net revenue on the walnut orchard (ignoring common costs listed in the introduction) in year t given the orchard is in its s th year of operation is equal to, where ܴ ௐ௧ is the walnut tree removal cost in year t (ܴ ௐ௧ = 0 in all years except a walnut orchard's last) and ‫ܭ‬ ௐ௧ is the year t annual pump/well and irrigation equipment cost on a walnut orchard and the remaining acres of wheat. ‫ܭ‬ ௐ௧ needs to be positive every year a walnut orchard is on the land. Further, ܻ ௐ௧(௦) and ‫ܥ‬ ௐ௧ሺ௦ሻ could equal 0 but ‫ܭ‬ ௐ௧ > 0 if the farm is no longer in walnuts but the walnut well/pump and irrigation equipment is still being paid for.

Prune orchard
In California's central valley, a typical prune orchard covers 100 acres. A prune orchard does not produce any fruit until its fourth year and does not typically reach its maximum annual yield levels until its 7 th year. A typical prune orchard has a lifetime (rotation) of 30 years. If the farm remains in a prune orchard for 30 years then at the end of the 30 th year all the prune trees are ripped out of the ground. Then either new prune trees are planted or the farmer converts to a new land use.
The first year of a prune orchard is mainly site prep and tree planting (there is some tree planting in the second year as well). Pruning begins immediately. Bee hives are rented every year beginning in the fourth year. The farmer rents a custom harvester every year beginning in the fourth year. The farmer also pays for pitting and dehydrating the prunes at an offsite facility. Every year, fertilizer, various pesticides (herbicides, pesticides, and rodenticides), and irrigation water is applied to the orchard. A drip irrigation system is used on prune orchards.
Annual per acre operating costs in a prune orchard is given by, where Plant Pt ( Annual net operating revenue on the prune orchard in in year t given that it is in its s th year of operation is equal to, where ‫‬ ௧ is the market price for a (short) ton of dried prunes in year t, ܻ ௧(௦) is the (dry short) tons of prunes yielded per ace in year t given the orchard is s years old, and ‫ܥ‬ ௧(௦) (equation 7) is the orchard's per acre operating cost in year t given the orchard is s years old.
The capital costs unique to prune farming (and therefore not ignored) are the costs for a pump/well, drip irrigation system, and second tractor. While a prune orchard can be abandoned after 5, 10, 15, 20, or 25 years of operation we assume the loan taken out to build and maintain the pump/well, purchase the drip irrigation equipment, and purchase the second tractor at the time of orchard establishment is sunk (and the equipment cannot be sold to another farmer to partially recover costs).
We assume the pump/well and irrigation system covers 100 acres, has a 30 year lifespan, costs $290,000, and can be paid with a loan with an interest rate of 4.75. According to the Excel functions PMT(0.0475,25,290000) this means an annual payment of $18,331 per year. Because we assume no re-sale of irrigation systems the equipment is always available to the farmer for 30 years after purchase. Therefore, for example, if the farmer establishes a prune orchard in year t = 0, abandons it after 10 years, and then begins a prune orchard again at t = 19 she still has 10 years of a pump/well and drip irrigation equipment before she has to purchase and install a new irrigation system. The removal of prune trees from the farm at the end of the 30 year rotation or earlier if the farmer abandons prune farming for another use is also attributed to the prune operation in our model. According to the prune enterprise budget tree removal is $150 per acre.
Annual net revenue on the prune orchard (ignoring common costs) in year t given the orchard is in its s th year of operation is equal to, where ܴ ௧ is the prune tree removal cost in year t (ܴ ௧ = 0 in all years except a prune orchard's last) and ‫ܭ‬ ௧ is the year t pump/well, irrigation equipment, and additional tractor cost on the prune orchard. ‫ܭ‬ ௧ needs to be positive every year a prune orchard is on the land. Further, ܻ ௧(௦) and ‫ܥ‬ ௧ሺ௦ሻ could equal 0 but ‫ܭ‬ ௧ > 0 if the farm is no longer growing prunes but the prune irrigation equipment and additional tractor are still being paid for.

Pasture
If this Central Valley farm is converted to pasture for hay production then the farmer has to prep the land with clearing, chiseling, and disking. Then in the late summer of the first year seed is applied: 12 pounds of orchardgrass or 16 pounds of tall fescue seed per acre and two pounds of clover seed (ladino, alsike, strawberry, or white Dutch) per acre. A custom operator does the planting. Beginning in the second year, every June the grass is custom harvested. Other than renting the custom harvester, the famer is not responsible for any harvest costs.
Every year fertilizer, herbicides, rodenticide, and irrigation water are applied to the pasture. Border-flood irrigation is used on pasture. A typical grass stand has a lifetime of 20 years. Therefore, the 21 st, 41 st , 61 st , and 81 st year of a pasture (s = 21, 41, 61, 81) are establishment years less the land preparation costs, clearing, chiseling, and disking, which are only necessary when the pasture land use is established.
Therefore, each year the operating costs in a pasture is given by, where Prep Ht ( Carbon Ht(s) is the per acre tax on net carbon emissions from a pasture in year t given the pasture is in its s th year. Any paid labor and machine repair costs are included in these variables.
Annual net operating revenue on the pasture in year t given that it is in its s th year of operation is equal to, where ‫‬ ு௧ is the market price for a (short) ton of hay in year t, ܻ ு௧(௦) is the (short) tons of hay yielded per acre in year t given the pasture is s years old, and ‫ܥ‬ ு௧(௦) (equation 10) is the pasture's per acre operating cost in year t given the pasture is s years old. We assume that cattle are never grazed on the pasture and therefore no electric fencing is needed (this is a cost in the enterprise budget).
The two capital costs that are unique to hay farming (and therefore not ignored) are the flood irrigation system and the hay barn. While a pasture can be abandoned at any five year increment we assume the loan taken out to build and maintain the irrigation pump and well and build a hay barn at the time of pasture establishment or every 25 years hence is sunk (and the equipment cannot be sold to another farmer to partially recover costs). We assume the flood irrigation system covers 100 acres, has a 25 year lifespan, costs $15,750, and a loan interest rate of 4.75%. We assume the hay barn has a 25 year lifespan, costs $50,638, and a loan interest rate of 4.75%. According to the Excel equations PMT(0.0475,25,15750) and PMT(0.0475,25,50,638) this means annual payments of $1,090 and $3,503 for the irrigation equipment and hay barn, respectively. Because we assume no re-sale of irrigation systems the equipment is always available to the farmer for 25 years after purchase. Therefore, for example, if the farmer establishes a pasture in year t = 0, abandons it after 10 years, and then begins a pasture again at t = 19 she still has 5 years of irrigation equipment and a hay barn before she has to purchase and install a new irrigation system and build a new hay barn.
Therefore, annual net revenue on the pasture (ignoring common costs) in year t given the orchard is in its s th year of operation is equal to, where ‫ܭ‬ ு௧ is the year t irrigation equipment and hay barn cost on a pasture. ‫ܭ‬ ு௧ needs to be positive every year the land is in pasture. Further, ܻ ு௧(௦) and ‫ܥ‬ ு௧ሺ௦ሻ could equal 0 but ‫ܭ‬ ு௧ > 0 if the farm is no longer in pasture but the pasture irrigation equipment and hay barn are still being paid for.

Wheat
Almond and walnut orchards do not cover 100 acres. Therefore, when the farm is used for almond or walnut production the farmer needs to place the remaining acres in some other productive use. Here we use the net returns to irrigated wheat to represent the value of that other productive use. In this analysis we assume continuous wheat (i.e., wheat is produced on these remaining acres every year that almonds or walnuts are produced on the farm). In reality such a choice may not be advisable; for example, it may make more financial sense for the wheat to be rotated with tomatoes or for the wheat acres to be left fallow every 5 th or 10 th year. To make continuous wheat a bit more realistic we assume that the farmer always leaves all wheat straw on the field to return nutrients to the soil. 18 Here we use the enterprise budget for irrigated wheat grown in the southern half of the San Joaquin valley. The production year for wheat begins in the fall. In November the wheat acres are disked, fertilized, and seeded. In the spring additional fertilizer is combined with irrigation water. Herbicide is applied in January and harvest occurs in June. A custom operator does the harvesting and mills pay the hauling costs from the field.
Therefore, each year the operating costs in an acre of wheat is, where Prep Ct is the per acre cost of prepping the land for wheat production each year, Plant Ct is the per acre cost of planting wheat each year, Pest Ct is the annual per acre pesticide cost in the wheat field, Fert Ct is the annual per acre fertilizer cost in the wheat field, IrrWater Ct is the annual per acre irrigation cost (the cost of water) in the wheat field, VehCost Ct is the per acre cost of operating a truck and ATV on the wheat field in year t, Interest Ct is the per acre interest payment on operating capital cost used on a wheat field in year t , 19 Harvest Ct is the annual per acre harvest cost in the wheat field, and Carbon Ct is the per acre tax on net carbon emissions from a wheat field in year t. Any paid labor costs are included in these variables.
Annual net operating revenue from wheat in year t is equal to, or where ‫‬ ௧ is the market price for a ton of wheat in year t, ܻ ௧ is the tons of wheat yielded per acre in year t, and ‫ܥ‬ ௧ (equation 13) is the wheat field's per acre operating cost in year t. Equation (14) is used if the remaining acres are in almonds and equation (15) is used if the farm's remaining acres are in walnuts.
We have already accounted for the capital costs of the wheat irrigation system in the almond and walnut net return calculations. The two capital costs that are unique to wheat farming (and therefore not ignored) are two additional tractors, two diskers, and a planter drill. However, the budget sheet assumes a 600 acre farm. Given the small nature of our farm we assume the need for only one additional tractor, disker, and planter. Therefore, annual net revenue on the wheat farm (ignoring common costs) in year t is equal to, or where ‫ܭ‬ ௧ is the year t tractor, disker, and planter cost on the wheat farm. ‫ܭ‬ ௧ needs to be positive every year an almond or walnut orchard is on the land. Further, in some years ܻ ௧ and ‫ܥ‬ ௧ could equal 0 but ‫ܭ‬ ௧ > 0 if the farm is no longer in almonds or walnuts but the wheat farming equipment is still being paid for.

Crop prices
To model future crop prices (i.e., ‫‬ ௧ , ‫‬ ௐ௧ , ‫‬ ௧ , ‫‬ ு௧ , and ‫‬ ௧ ) we first collected historical price information from QuickStats on the USDA-NASS website. All prices are specific to California.  and the covariance-variance matrix given in Table 3. We project real prices (2011 $) for the years t = 1,…,99 for each crop j with a random walk where, where ‫‬ is given by j's price in the mean vector p and ‫ݎ‬ ௧ is taken from the a 5 element vector randomly drawn from the multivariate normal distribution with mean [0 0 0 0 0] and a covariance-variance matrix as given in Table 5. A new random draw occurs at each t. If ‫‬ ௧ < 0 then ‫‬ ௧ = ‫‬ ௧ିଵ .

Net GHG emissions from orchard or pasture use
Annual net emissions on an orchard are measured in Mg per acre per year. They come from Marvinney et al. These values are used to determine the annual farm-level cost of a carbon tax .

Net operating revenue matrices
For each crop j we create a 20 x 20 net operating revenue matrix. Let this matrix be given by NR j . The first row of the matrix corresponds to the years t = 0,…,4, the second row to the years t = 5,…,9, the third row to the years t = 10,…,14, etc. The columns of the matrix j correspond to the number of consecutive 5-year increments or blocks that the land has been in land use j. Finally, the element of matrix j gives the present value of the sum of net operating revenues over the years t,…,t+4 for a land use that is in its c th consecutive 5-year increment. The upper triangle of NR j contains all 0s.
To illustrate NR j consider an almond orchard (and wheat on the remaining acres) established at the beginning of year t = 0. The present value of the net operating revenue generated by such an operation over its first 5 years (t = 0,…,4 and s = 1,…,5) is given at matrix element 1,1 and is equal to, The present value of the net operating revenue generated by this farm in its 6 th consecutive 5 year increment or block of almond farming (t = 25,…,29 and s = 1,…,5) is given at element {6,6} and is equal to, Recall that for almond and wheat orchards that have been on the farm for 25, 50, or 75 consecutive years beginning another 5-year stint in the same orchard type means a new orchard rotation and the age of the orchard during its 6 th consecutive 5 year increment or block of almond farming is given by s = 1,…,5.
Recall that the 4 th column means the 16 th through 20 th consecutive years of an almond farm and therefore s = 16,…,20 for at matrix position(6,4).
The process of creating ‫܀ۼ‬ ௐ is identical to the process of creating ‫܀ۼ‬ (however, the acre multiplier attached to the sum of net operating revenues from the orchard and the wheat farm will be different). The creation of ‫܀ۼ‬ is very similar except a new rotation begins after the 30 th , 60 th , and 95 th consecutive years of a prune orchard and there are no wheat returns to include. Finally, when creating ‫܀ۼ‬ ு , the net operating revenue matrix for pasture, a new stand has to be established after 20, 40, 60, and 80 consecutive years in pasture.

MATLAB Code for orchard and pasture use
The variance-covariance matrix in Table 3  Every time we run the model we generate a new 100 year price matrix.

Perpetual conservation easement of forest cover
We assume a conservation easement of reforested cover on the farm is irreversible. Assume the farmer chooses to establish the reforested easement in year t. Therefore, year t is the "date of offset project commencement" (DOPC) or the year trees are planted across the entire 100 acres of the farm (see section 3.2 of the Protocol). According to the Compliance Offset Protocol for Forest Projects, the offset project must be listed with authorities within 6 months of DOPC (see section 3.3 of the Protocol). For simplicity we assume both the DOPC and listing occurs in year t. The listing must include 1) details on the inventory sampling methodology that will be used in future years to quantify carbon storage, 2) models that will be used to estimate the baseline carbon trajectory, 3) models that will be used to quantify the expected carbon sequestration trajectory given the easement, and 4) plans for an annual monitoring of the project. According to the Protocol, the Project's life begins with the DOPC and ends with the year carbon credits are first issued plus 100 years.
Let the per acre cost of starting up the carbon project and completing the listing be given by L t . According to Saah et al. (2014) the cost of starting the carbon project and completing the listing over a 100 acre plot is $186 per acre. 20 The cost of planting the forest is given by Pl t . Let the one time per acre payment to the farmer for accepting the easement in year t be given by E t. The easement is paid by a local land trust and by default is equal to 50% of the land's assessed value. The other first year payment to the farmer for converting to perpetual forest use, giving by Sal t , is the salvage value from selling an unnecessary tractor and fuel tank The aforementioned baseline carbon trajectory gives the expected carbon storage values at the end of years t, t + 1, t + 2, t + 3, … if the farm was left in some baseline agricultural use instead of a conservation easement. The series BC t , BC t+1 , BC t+2 , BC t+3 ,… has to be quantified in the initial listing. In our model the baseline for a newly established forest is set by the land use and the age of the land use in the previous time step. For example, if the land use immediately previous to forest establishment was an almond orchard in its 15 th through 20 th years of rotation (s = 15 -20) then BC t would indicate the amount of woody biomass carbon stored in on the farm at the end of the orchard's 21 st year, BC t+1 would indicate the amount of woody biomass carbon stored on the farm at the end of the orchard's 22 nd year, etc. The series BC t , BC t+1 , BC t+2 , BC t+3 ,… would continue to assume an almond orchard is re-established on the farm at the end of orchard's 25 th year in rotation (s = 25). If forest use begins immediately at t = 0 the baseline is an almond orchard in its first year of rotation. Further, as noted above, the listing also must contain an estimate of C t , C t+1 , C t+2 , C t+3 , …, the expected amount of carbon stored in the woody biomass of the reforested farm at the end of years t, t + 1, t + 2, t + 3,… We assume that an acre of almond orchard sequesters 0.176 Mg of C per year in woody biomass and an acre of walnut and prune orchard sequester 0.252 Mg of C per year in woody biomass. Finally, we assume an acre of wheat and pasture has no woody biomass. This carbon accumulates in a linear fashion until the orchard has to be cleared and reestablished. For example, suppose in year t an almond orchard has completed its 25 th year of rotation. At this point the 100 acre farm will have accumulated 0.176 x 40 x 25 = 176 Mg of C (the 40 reflects the fact that only 40 acres of the farm will be in almond orchard). Between years t and t + 1 the orchard will be cleared and a new almond orchard will be established BC t+1 -BC t = 7 -176 = -169 Mg of C where the 7 Mg of C reflects the sequestration in the first year of newly established almond orchard.
Before receiving any credits for "additional" carbon sequestration (adjusted C less BC) from the state of California, the farm must undergo an initial verification. If the DOPC occurs in year t then the initial verification occurs at the end of year t + v. The value for v can either set to 3, 9, 15, or 21 (the 4 th , 10 th , 16 th , or 22 nd year of the forest). After initial verification the project is eligible to receive Offset credits as long as the afforestation project meets program requirements.
During the initial verification at the end of year t + v the carbon stored in the woody biomass on the farm is determined with a plot sampling procedure. The California protocol lets the farmer choose the intensity of plot sampling. If the farmer chooses 'high' then the sampling procedure is very detailed, if the farmer chooses 'medium' then the sampling is less detailed, and if the farmer chooses 'low' then the sampling is the minimum level of sampling intensity necessary to qualify for any carbon credits. From this sampling the amount of carbon stored on the farm is determined and the modeled series C t , C t+1 , C t+2 , C t+3 , …, C t+v is updated accordingly. As long as the farmer chooses high intensity sampling (and earns a Confidence Deduction (CD) equal to 0) then the estimated series C t , C t+1 , C t+2 , C t+3 , …, C t+v does not have to be adjusted. If the farmer chooses medium or low intensity sampling then all storage values have to be reduced 12% or 20%, respectively. The CD adjustment is meant to give the famer financial incentive to be as accurate as possible in her assessment of the easement's woody biomass carbon storage. We assume the farmer chooses a high intensity sampling and earns a CD of 0. The per acre cost of the initial verification, IV t+v , increase in v.
According to the Protocol, a second verification must occur 12 years after the initial verification and subsequent verifications every 6 years thereafter until the end of the project's life. In each subsequent verification year the intensity of sampling is chosen by the farmer and the CDadjusted woody biomass storage values for the years since the previous verification are calculated accordingly. Again we assume the farmer always chooses a CD of 0 and does not have to adjust her estimated carbon storage down. The per acre costs of the 6-year and 12-year verification are given by SV t+v+18 and TV t+v+12 . From this sampling the amount of carbon stored on the farm is determined and the modeled series C t , C t+1 , C t+2 , C t+3 , …, C t+v+18 is updated accordingly.
Per acre verification costs assuming a CD of 0 are given in the table below. The modeler chooses the initial verification period; the timing of all subsequent verifications is based on the choice of v. Carbon credits can only be created with certain forest types. The reforestation project must include species native to the ecoregion "California Central Valley Basin, California Valley Oak Woodland." Such species include blue oak, California black oak, California laurel, coast live oak, gray pine, pinyon, juniper, knobcone pine, pacific madrone, canyon live oak/interior live oak, western oak, cottonwood, and willow. Further, the species diversity index must be at least 65%. Here we consider 3 different forest stands that meet this requirement. The first is a Western Oak stand as described in Smith et al. (2006). The second is a Mixed Conifer stands as described in Smith et al. (2006). The third is a Mixed Riparian as described in Matzek et al. (2014).
The carbon stored in each stand is given in Table 5. From these data create the C t , C t+1 , C t+2 , C t+3 , …, C t+v trajectory used in the model. The modeler chooses which trajectory the model uses.

Quantification of GHG removals
According to the Protocol, the metric tons of carbon removed from the atmosphere at the end of year t by the reforested farm is given by, where t is the years the forest is established, z = 0 at the end of the forest's first year, z = 1 at the end of the forest's second year, etc., ߚ ϵ [0,1) is the assumed "leakage risk" associated with the reforestation project, C and BC are defined above, ‫ܦܥ‬ ௧ିଵ , ‫ܥ‬ ௧ିଵ , ‫ܥܤ‬ ௧ିଵ , and ܴܳ ௧ିଵ are all equal to 0, MC is the carbon emitted during the establishment of the forest, and ‫ܥܯ‬ ௧ା௭ = 0 for all z ≥ 1. By default we assume ߚ = 0.24.
Therefore, ܴܳ ௧ is given by, at the end of the forest's first year (z = 0) and by at the end of all other years in the life of the forest (z ≥ 0). Given that carbon storage values are measured at the end of the year, QR t+z -QR t+z-1 measures the amount of "additional" sequestered from January 1 to December 31 of year t + z. For example, suppose the forest was established in t = 50. QR 50+0 -0 gives additional carbon sequestered from January 1 to December 31 in t = 50. Further, QR 50+1 -QR 50+0 gives additional carbon sequestered from January 1 to December 31 in t = 51. Further, QR 50+2 -QR 50+1 gives additional carbon sequestered from January 1 to December 31 of year 52, etc. The farm earns carbon credits at the end of year t + z if 1) ܴܳ ௧ା௭ > 0 and 2) year t + z is a verification year (i.e., t + z = t + v; t + z = t + v + 12; or t + z = t + v + 18; etc.). A positive ܴܳ ௧ା௭ score is converted into carbon permits at the end of year t + z with the following formula, where ‫ܨܴ‬ ௧ା௭ is the cumulative risk factor of partial or full forest destruction in year t + z. ‫ܨܴ‬ ௧ା௭ is given by, where ߛ ௧ା௭ is the financial failure risk in year t + z, ߜ ௧ା௭ is the illegal forest biomass removal risk in year t + z, ߠ ௧ା௭ is the land-use conversion risk in year t + z, ߤ ௧ା௭ is the overharvesting risk in year t + z, ߪ ௧ା௭ is the social risk in year t + z, ߬ ௧ା௭ is the wildfire risk in year t + z, ߮ ௧ା௭ is the disease and insect outbreak risk in year t + z, and ߱ ௧ା௭ is other catastrophic event risk in year t + z. All risks range from 0 to 1 where a lower value means that the particular risk to forest biomass in year t + z is deemed to be small. If all risk factors are equal to 0 then ‫ܨܴ‬ ௧ା௭ = 1 and ‫ܲܥ‬ ௧ା௭ = ܴܳ ௧ା௭ ሺ1 − 0ሻ = ܴܳ ௧ା௭ . In other words, the famer is paid for all adjusted "additional" sequestration created at the end of verification year t + s. If ‫ܨܴ‬ ௧ା௭ < 1 then ‫ܲܥ‬ ௧ା௭ < ܴܳ ௧ା௭ and the farmer is obliged to place ܴܳ ௧ା௭ − ‫ܲܥ‬ ௧ା௭ credits in a bank that will be used if biomass is unintentionally destroyed on the farm in the future.
The Protocol sets the fire risk rating for forests with no fuel management regime in this part of California, at ߬ ௧ା௭ = 0.04 for all t + z. Because we assume lands are covered by a qualified conservation easement, the default risk for ߛ ௧ା௭ is 0.01 for all t + z, the default risk for ߠ ௧ା௭ is 0 for all t + z, and the default risk for ߤ ௧ା௭ is 0 for all t + z. The other risk factors are set by default for any U.S. forest project at these values: ߜ ௧ା௭ is 0 for all t + z, ߪ ௧ା௭ is 0.02 for all t + z, ߮ ௧ା௦ is 0.03 for all t + z, and ߱ ௧ା௭ is 0.03 for all t + z. Therefore, equation (27)

Forest net revenue matrices
We create a 20 x 20 x 4 x 6 net revenue matrix for forest use. Let this matrix be given by NR f . The first row of the matrix corresponds to the years t = 0,…,4, the second row to the years t = 5,…,9, the third row to the years t = 10,…,14, etc. The columns of the matrix j correspond to the number of consecutive 5-year increments that the land has been in forest. The third dimension of the matrix indicates the farm's land use in t -1 (almond orchard (1), walnut orchard (2), prune orchard (3), or pasture (4) where ‫‬ ௧ is the real price of a carbon permit (an adjusted metric ton of carbon) at the end of year t, tr t = 0.48 is the transition cost per permit (or metric ton) in year t, and ‫ܲܥ‬ ௧ା௩ (: , : ,1,1) gives the carbon credits issued in verification year v. ‫ܲܥ‬ ௧ା௩ is indexed by the last 2 dimensions because it will be a function of the previous land use and the counterfactual rotation age of the previous use. The matrix element ۱ (1,1,1,1) is given by, where the cost of an annual report, given by AR t , is $20 per acre. o The matrix dnrAlmCredits is a 120 x 20 matrix of the net returns to carbon farming assuming a baseline land use and baseline. Each block of 20 rows assumes a different baseline land use year. For example, the first 20 rows of 'dnrAlmCredits' assume the baseline almond orchard had just been established. The second block of 20 rows assumes the baseline almond orchard would have begun its 6 th year when the forest was established. The third block of 20 rows assumes the baseline almond orchard would have begun its 16 th year when the forest was established. The fourth block of 20 rows assumes the baseline almond orchard would have begun its 20 th year when the forest was established. The fifth block of 20 is empty. Within a block of 20 each row is a 5-year time step in the 100-year simulation and each column is a 5-year time step in the life of a carbon farm. Each cell is the sum of discounted net returns over the given 5-year period. o The matrix dnrWalCredits is similar to dnrAlmCredits o The matrix dnrPruCredits has data in all 5 blocks of 20. The fifth block of 20 rows assumes the baseline prune orchard would have begun its 25 th year when the forest was established. o The matrix dnrPasCredits only has data in the first block of 20 because baseline year is irrelevant for pasture.

Land use trajectories
We create a matrix with 1,725,961 land use trajectories for the years t = 0,…,99 with the constraint that land use can only be changed every 5 th year and the conversion to forest is irreversible. For example, the land use vector, indicates that that farm contained 40 acres of almond orchard (and 60 acres of wheat) in years t = 0-14, 60 acres of walnut orchard (and 40 acres of wheat) in years t = 15-39, and in pasture for the years t = 40 -99. Or the land use vector, [1 1 1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1] indicates that that farm contained 40 acres of almond orchard (and 60 acres of wheat) in years t = 0-14, 60 acres of walnut orchard (and 40 acres of wheat) in years t = 15-49, and in almonds (and 60 acres of wheat) for the years t = 50 -99. Finally, the land use vector, [1 1 1 2 2 2 2 2 2 2 12 12 12 12 12 12 12 12 12 12] indicates that that farm contained 40 acres of almond orchard (and 60 acres of wheat) in years t = 0-14, 60 acres of walnut orchard (and 40 acres of wheat) in years t = 15-49, and forest the years t = 50 -99. Any land code of 5 or greater indicates forest. In this case the 12 indicates that the previous land use was a walnut farm that had just finished its second block of a 5 year rotation period. In other words, if the walnut orchard had continued it would have begun its 11 th year of rotation in the year the forest was established. Any 21 Accompanying the land us trajectory matrix is a timing matrix that indicates the consecutive number of 5 year blocks that the farm has been on the same land use. For example, the timing vectors that corresponds to vectors (39) -(41) would be equal to, [1 2 3 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 11 12] [1 2 3 1 2 3 4 5 6 7 1 2 3 1 2 3 4 5 6 7 8 9 10] [1 2 3 1 2 3 4 5 6 7 1 2 3 1 2 3 4 5 6 7 8 9 10] When a land use vector switches to forest the land code is assigned by evaluating the previous 5-year block's land use code and rotation time.
14. Running the model For a given a 100 year price matrix, the model evaluates each of the 1,725,961 land use trajectory. Specifically, each element in the land use trajectory matrix is assigned a value from one of the NR matrices according to the element's land use and rotation year. Subsequently, sunk costs and orchard clearing costs are added to each element accordingly. The trajectory that returns the greatest present value of net returns solves the optimization model.
Every time we run the model we generate a new 100 year price matrix.