Appendix to “Carbon credits compete poorly with agricultural commodities in an optimized
model of land use in Northern California”
Erik Nelson1,2 and Virginia Matzek3
Assume a farmer of 100 acres whose goal is to maximize the present value of net returns to her
land over the next 100 years. Over the course of the 100 years the farm is always in one of the
following land uses,
1.
2.
3.
4.
5.
almonds (40 acres) and wheat (60) acres;4
walnuts (60 acres) and wheat (40) acres;5
prunes (100 acres);
pasture (100 acres); or
perpetual conservation easement of forest cover (100 acres).
Let discrete time be indexed by t = 0,…,99. To make the model more tractable we assume that
a farmer will only consider changing her land every 5 years. So, for example, the farm will be in
its initial land use for the time period t = 0,…,4. Then beginning at t = 5 the farm can be in a
new use or in the previous use. This subsequent decision lasts through t = 9 and then a new
decision has to be made for the period t = 10,…,14, etc. We assume that the decision to use the
land for perpetual forest cover is irreversible. All other land use decisions are reversible.
Next we discuss in detail the net returns to each land use. Please note that we ignore the costs
that are common to all land uses. We do this because we are interested in finding the land use
trajectory that maximizes the present value of net returns to the farm and not necessarily the
exact net returns to her land. Costs that are common across all uses will not affect her
decisions and thus can be dropped for expediency’s sake. Therefore, annual operational,
repair, and capital costs associated with 1) land; 2) two fuel tanks;6 3) shop and field tools; 4)
miscellaneous farm equipment (includes mower, weed sprayer, harrow, etc.); 5) a 2400 square
foot building (some land uses require additional buildings; those costs are included in our
analysis); 6) tractor (some land uses require additional tractors; those costs are included in our
analysis);7 and 7) cash overhead costs (includes office expenses, liability insurance, sanitation
fees, environmental fees, regulatory fees, property taxes, property insurance, farm manager
salary, and investment repairs). We also ignore the capital costs associated with an assumed
all-terrain vehicle (ATV) and truck (some land uses require additional trucks; those costs are
included in our analysis). We do include the ATV and truck’s operation costs as those vary
according to land use.
1
Department of Economics, Bowdoin College, Maine, USA.
Centre for Environmental and Climate Research, Ecology Building, Lund University, S-223 62 Lund, Sweden.
3
Department of Environmental Studies and Sciences, Santa Clara University, California, USA
4
This is the acre mix assumed by the U of CA – Davis budget sheet for almonds.
5
This is the acre mix assumed by the U of CA – Davis budget sheet for walnuts.
6
We do assume a farmer that adopts forest cover will immediately salvage their fuel tanks.
7
We do assume a farmer that adopts forest cover will immediately salvage the tractor.
2
1
1. 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 6th 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 25th 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,
஺௧(௦) = ஺௧(௦) + ஺௧(௦) + ஺௧(௦) + ஺௧(௦) +
஺௧ሺ௦ሻ + ஺௧(௦) + ஺௧(௦) + ℎ஺௧(௦) +
஺௧(௦) + ஺௧(௦) + ஺௧(௦)
(1)
where PlantAt(s) is the per acre orchard planting and site prep cost in year t given the orchard is
in its sth year (PlantAt(s) = 0 for s ≥ 3), PestAt(s) is the per acre orchard pesticide cost in year t given
the orchard is in its sth year, FertAt(s) is the per acre orchard fertilizer cost in the year t given the
orchard is in its sth year, IrrWaterAt(s) is the per acre irrigation water cost in year t given the
orchard is in its sth year,8 PruneAt(s) is the per acre orchard pruning cost in year t given the
orchard is in its sth year (PruneAt(s) = 0 for s ≤ 2), PollinAt(s) is the per acre cost of renting beehives
in year t given the orchard is in its sth year (PollinAt(s) = 0 for s ≤ 2), SanitationAt(s) is the per acre
cost of removing mummy nuts from trees in year t given the orchard stand is in its sth year
(SanitationAt(s) = 0 for s ≤ 2), VehCostAt(s) is the per acre cost of operating a Truck and ATV in year
t given the orchard is in its sth year, InterestAt(s) is the per acre interest payment on operating
capital cost in year t given the orchard is in its sth year,9 HarvestAt(s) is the per acre orchard
harvest cost in year t given the orchard is in its sth year (HarvestAt(s) = 0 for s ≤ 2), and CarbonAt(s)
is the per acre tax on net carbon emissions from an almond orchard in year t given the orchard
is in its sth year. The value of s cannot exceed 25. Any paid labor and equipment repair costs are
included in these variables.
8
This is just the cost of water, the labor associated with irrigation, and any repair costs. The capital costs for the
irrigation equipment are added in later.
9
Assumes a 5.75% per annum rate.
2
Annual net operating revenue on the almond orchard in year t in its sth year of rotation is equal
to,
஺௧(௦) = 40஺௧ ஺௧(௦) − ஺௧(௦) (2)
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 ஺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 acres10, 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 sth year of operation is equal to,
஺௧(௦) = 40஺௧ ஺௧(௦) − ஺௧ሺ௦ሻ − ஺௧ − ஺௧
(3)
where ஺௧ is the almond tree removal cost in year t (஺௧ = 0 in all years except an almond
orchard’s last) and ஺௧ is the year t irrigation equipment cost on an almond orchard and the
remaining wheat acres. ஺௧ needs to be positive every year an almond orchard is on the land.
Further, in some years ஺௧(௦) and ஺௧ሺ௦ሻ could equal 0 but ஺௧(௦) > 0 if the farm is no longer in
almond production but the almond irrigation equipment loan is still being paid off.
2. 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 8th year. A typical
10
The budget sheet assumes the irrigation equipment covers 95 acres, we prorate costs to 100 acres by
multiplying capital costs by 100/95.
3
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 25th 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,
ௐ௧(௦) = ௐ௧(௦) + ௐ௧(௦) + ௐ௧(௦) +
ௐ௧(௦) + ௐ௧(௦) + ℎ஺௧(௦) +
ௐ௧(௦) + ௐ௧(௦) + ௐ௧(௦)
(4)
where PlantWt(s) is the per acre orchard planting and site prep cost in the sth year of the orchard
(PlantWt(s) = 0 for s ≥ 3), PestWt(s) is the per acre orchard pesticide cost in the sth year of the
orchard, FertWt(s) is the per acre orchard fertilizer cost in the sth year of the orchard, IrrWaterWt(s)
is the per acre irrigation water cost in the sth year of the orchard, PruneWt(s) is the per acre
orchard pruning cost in the sth year of the orchard, VehCostWt(s) is the per acre cost of operating
a Truck and ATV in year t given the orchard is in its sth year, InterestWt(s) is the per acre interest
payment on operating capital cost in year t given the orchard is in its sth year,11 HarvestWt(s) is
the per acre orchard harvest cost in the sth year of the orchard (HarvestWt(s) = 0 for s ≤ 3)12,
CarbonWt(s) is the per acre tax on net carbon emissions from a walnut orchard in year t given the
orchard is in its sth 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 sth year of
operation is equal to,
ௐ௧(௦) = 60ௐ௧ ௐ௧(௦) − ௐ௧(௦) (5)
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
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,
11
12
Assumes a 5.75% per annum rate.
The walnut harvest cost includes a $0.01 / pound of yield payment to the CA Walnut Commission.
4
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 acres13, 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 sth year of operation is equal to,
ௐ௧(௦) = 60ௐ௧ ௐ௧(௦) − ௐ௧ሺ௦ሻ − ௐ௧ − ௐ௧
(6)
where ௐ௧ is the walnut tree removal cost in year t (ௐ௧ = 0 in all years except a walnut
orchard’s last) and ௐ௧ is the year t annual pump/well and irrigation equipment cost on a
walnut orchard and the remaining acres of wheat. ௐ௧ needs to be positive every year a
walnut orchard is on the land. Further, ௐ௧(௦) and ௐ௧ሺ௦ሻ could equal 0 but ௐ௧ > 0 if the farm is
no longer in walnuts but the walnut well/pump and irrigation equipment is still being paid for.
3. 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 7th 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 30th 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
13
The budget sheet assumes 60 acres, we prorate costs to 100 acres by multiplying capital costs by 100/60.
5
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,
௉௧(௦) = ௉௧(௦) + ௉௧(௦) + ௉௧(௦) + ௉௧(௦) +
௉௧(௦) + ௉௧(௦) + ௉௧(௦) + ℎ௉௧(௦)
+
௉௧(௦) + ௉௧(௦) + ௉௧(௦)
(7)
where PlantPt(s) is the per acre orchard planting and site prep cost in the sth year of the orchard
(PlantPt(s) = 0 for s ≥ 3), PestPt(s) is the per acre orchard pesticide cost in the sth year of the
orchard, FertPt(s) is the per acre orchard fertilizer cost in the sth year of the orchard, IrrWaterPt(s)
is the per acre irrigation water cost in the sth year of the orchard, PrunePt(s) is the per acre
orchard pruning cost in the sth year of the orchard, PollinPt(s) is per acre cost of renting beehives
in the sth year of the orchard (PollinPt(s) = 0 for s ≤ 3), ConsultationPt(s) is the per acre prune
marketing fee in the sth year of the orchard (ConsultationPt(s) = 0 for s ≤ 3),14 VehCostPt(s) is the
per acre cost of operating a Truck and ATV in year t given the orchard is in its sth year,
InterestPt(s) is the per acre interest payment on operating capital cost in year t given the orchard
is in its sth year,15 HarvestPt(s) is the per acre harvest cost in the sth year of the orchard
(HarvestPt(s) = 0 for s ≤ 3), 16 CarbonPt(s) is the per acre tax on net carbon emissions from a prune
orchard in year t given the orchard is in its sth year. s cannot exceed 30. Any paid labor and
equipment repair costs are included in these variables.
Annual net operating revenue on the prune orchard in in year t given that it is in its sth year of
operation is equal to,
௉௧(௦) = 100௉௧ ௉௧(௦) − ௉௧(௦) (8)
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 ௉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).
14
Under a state marketing order, the California Prune Board collects mandatory assessment fees. This assessment
is charged to the grower to fund prune marketing, advertising, and research programs administered by the
California Prune Board.
15
Assumes a 5.75% per annum rate.
16
The prune harvest cost includes a $25.26 / dry ton of yield payment to the CA Prune Board.
6
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 second tractor, has a 15 year lifespan and has a price of $43,500 - $8,469 = $35,031 where
$8,469 is the assumed resale value at the end of 15 years. According to the Excel function
PMT(0.0475,15,35031) this means an annual payment of $3,318 per year for the tractor.
Because a prune orchard lasts 30 years the farmer will have to buy another tractor for s =
16,…,30. From the perspective of s = 1, the loan taken 15 years from then will have an annual
payment equal to PMT(0.0475,15,(35031/1.07^15)) or $1,203 per year for s = 16,…,30.
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 sth year of operation is equal to,
௉௧(௦) = 100௉௧ ௉௧(௦) − ௉௧ሺ௦ሻ − ௉௧ − ௉௧
(9)
where ௉௧ is the prune tree removal cost in year t (௉௧ = 0 in all years except a prune orchard’s
last) and ௉௧ is the year t pump/well, irrigation equipment, and additional tractor cost on the
prune orchard. ௉௧ needs to be positive every year a prune orchard is on the land. Further,
௉௧(௦) and ௉௧ሺ௦ሻ could equal 0 but ௉௧ > 0 if the farm is no longer growing prunes but the prune
irrigation equipment and additional tractor are still being paid for.
4. 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 21st, 41st, 61st, and 81st 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.
7
Therefore, each year the operating costs in a pasture is given by,
ு௧(௦) = ு௧(௦) + ு௧(௦) + ு௧(௦) + ு௧(௦) +
ு௧ሺ௦ሻ + ℎு௧ሺ௦ሻ + ு௧ሺ௦ሻ +
ு௧(௦) + ு௧(௦)
(10)
where PrepHt(s) is the per acre cost of prepping the land for hay production (Prep Ht(s) = 0 for s ≥
2), PlantHt(s) is the per acre grass planting cost in the sth year of the pasture (Plant Ht(s) > 0 for s =
1, 21, 41, 61, and 81), PestHt(s) is the per acre pesticide cost in the sth year of the pasture, FertHt(s)
is the per acre fertilizer cost in the sth year of the pasture, IrrWater Ht(s) is the per acre irrigation
water cost in the sth year of the pasture, VehCostHt(s) is the per acre cost of operating a truck
and ATV in year t given the pasture is in its sth year, InterestHt(s) is the per acre interest payment
on operating capital cost in year t given the pasture is in its sth year,17Harvest Ht(s) is the per acre
hay harvest cost in the sth year of the pasture (Harvest Ht(s) = 0 for s = 1, 21, 41, 61, and 81) and
CarbonHt(s) is the per acre tax on net carbon emissions from a pasture in year t given the pasture
is in its sth 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 sth year of operation
is equal to,
ு௧(௦) = 100ு௧ ு௧(௦) − ு௧(௦) (11)
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 ு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
17
Assumes a 5.75% per annum rate.
8
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 sth year of operation is equal to,
ு௧(௦) = 100ு௧ ு௧(௦) − ு௧ሺ௦ሻ − ு௧
(12)
where ு௧ is the year t irrigation equipment and hay barn cost on a pasture. ு௧ needs to be
positive every year the land is in pasture. Further, ு௧(௦) and ு௧ሺ௦ሻ could equal 0 but ு௧ > 0 if
the farm is no longer in pasture but the pasture irrigation equipment and hay barn are still
being paid for.
5. 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 5th or 10th
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,
஼௧ = ஼௧ + ஼௧ + ஼௧ + ஼௧ + ஼௧ +
ℎ஼௧ + ஼௧ + ஼௧ + ஼௧
(13)
where PrepCt is the per acre cost of prepping the land for wheat production each year, PlantCt is
the per acre cost of planting wheat each year, PestCt is the annual per acre pesticide cost in the
wheat field, FertCt is the annual per acre fertilizer cost in the wheat field, IrrWaterCt is the
annual per acre irrigation cost (the cost of water) in the wheat field, VehCostCt is the per acre
cost of operating a truck and ATV on the wheat field in year t, InterestCt is the per acre interest
18
http://www.ipni.net/ppiweb/bcrops.nsf/$webindex/9EC9DFE8E427AF508525762500659C77/$file/bc093p17.pdf
9
payment on operating capital cost used on a wheat field in year t ,19 HarvestCt is the annual per
acre harvest cost in the wheat field, and CarbonCt 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,
஼௧ = 100 − 40 ஼௧ ஼௧ − ஼௧ !
(14)
஼௧ = 100 − 60 ஼௧ ஼௧ − ஼௧ !
(15)
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 ஼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.
While a wheat farm can be abandoned at any five year increment we assume the loans taken
out to buy farm equipment for wheat farming are sunk (however, there is salvage value at the
end of the equipment’s run on the farm). The extra tractor has 10 year lifespan and price of
$114,269 - $33,753 = $80,516 where $33,753 is the expected salvage value. According to the
Excel function PMT(0.0475,10,80516) this means an annual payment of $10,301. The disker has
a 10 year lifespan and price of $2,150 - $380 = $1,770 where $380 is the expected salvage
value. According to the Excel function PMT(0.0475,10,1770) this means an annual payment of
$226.45. Finally, the planter drill has 10 year lifespan and price of $27,301 -$4,828 = $22,473
where $4,828 is the expected salvage value. According to the Excel function
PMT(0.0475,10,22473) this means an annual payment of $2,875 year.
Therefore, annual net revenue on the wheat farm (ignoring common costs) in year t is equal to,
஼௧ = 100 − 40 ஼௧ ஼௧ − ஼௧ ! − ஼௧
(16)
஼௧ = 100 − 60 ஼௧ ஼௧ − ஼௧ ! − ஼௧
(17)
or
19
Assumes a 5.75% per annum rate.
10
where ஼௧ is the year t tractor, disker, and planter cost on the wheat farm. ஼௧ needs to be
positive every year an almond or walnut orchard is on the land. Further, in some years ஼௧ and
஼௧ could equal 0 but ஼௧ > 0 if the farm is no longer in almonds or walnuts but the wheat
farming equipment is still being paid for.
6. Crop prices
To model future crop prices (i.e., ஺௧ , ௐ௧ , ௉௧ , ு௧ , and ஼௧ ) we first collected historical price
information from QuickStats on the USDA-NASS website. All prices are specific to California.
Table 1: Nominal market prices from California
Almonds ($ Walnuts ($
Prunes ($ per
Hay ($ per
Wheat ($ per
CPI
per pound) per pound)
dried short ton) short ton)
bushel)
2001
0.91
0.56
726
93.5
2.72
79
2002
1.11
0.585
810
79.5
3.41
80
2003
1.57
0.58
772
77
3.27
82
2004
2.21
0.695
1,500
95.5
3.32
84
2005
2.81
0.785
1,470
99.5
3.32
87
2006
2.06
0.815
1,390
89
4.17
90
2007
1.75
1.145
1,450
120
6.13
92
2008
1.45
0.64
1,500
148
6.57
96
2009
1.65
0.855
1,230
94
4.71
95
2010
1.79
1.02
1,350
97
5.37
97
2011
1.99
1.45
1,310
174
6.81
100
2012
2.58
1.28
1,250
183
7.55
102
Table 2: Real market prices from California (2011 = 100)
Almonds ($ Walnuts ($
Prunes ($ per
Hay ($ per
Wheat ($ per
bushel)
per pound) per pound)
dried short ton) short ton)
2001
1.15
0.71
918.99
118.35
3.44
2002
1.39
0.73
1012.50
99.38
4.26
2003
1.91
0.71
941.46
93.90
3.99
2004
2.63
0.83
1785.71
113.69
3.95
2005
3.23
0.90
1689.66
114.37
3.82
2006
2.29
0.91
1544.44
98.89
4.63
2007
1.90
1.24
1576.09
130.43
6.66
2008
1.51
0.67
1562.50
154.17
6.84
2009
1.74
0.90
1294.74
98.95
4.96
2010
1.85
1.05
1391.75
100.00
5.54
2011
1.99
1.45
1310.00
174.00
6.81
2012
2.53
1.25
1225.49
179.41
7.40
11
When we fit the real prices data from Table 2 to a multivariate normal distribution we get the
following mean vector p,
[஺ ௐ ௉ ு ஼ ] = [2.010 0.946 1354.44 122.96 5.19]
and the covariance- variance matrix given in Table 3.
Table 3: Price covariance- variance matrix
Almonds Walnuts Prunes
Hay
Wheat
Almonds
0.309
0.030
0.672
0.815 -0.036
Walnuts
0.030
0.055
7.049
3.895 0.211
Prunes
0.672
7.049 81562.82 520.63 3.225
Hay
0.815
3.895 520.630 794.410 28.65
Wheat
-0.036
0.211
3.225 28.653
1.77
We project real prices (2011 $) for the years t = 1,…,99 for each crop j with a random walk
where,
௝௧ = ௝௧ିଵ + ௝௧ for t = 1,…,99
(18)
where ௝଴ is given by j’s price in the mean vector p and ௝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.
7. 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.
Table 3: Net GHG emissions
Crop
Mg acre-1 year-1
Almond
0.1955
Walnut
0.2480
Prune
0.2480
Pasture
0.0937
Wheat
0.05144
These values are used to determine the annual farm-level cost of a carbon tax .
8. Net operating revenue matrices
For each crop j we create a 20 x 20 net operating revenue matrix. Let this matrix be given by
NRj. 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
12
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 cth consecutive 5-year increment. The upper
triangle of NRj contains all 0s.
To illustrate NRj 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,
"#஺ (1,1) = 40 ∑ସ௧ୀ଴
௣ಲ೟ ௒ಲ೟(೟శభ) ି஼ಲ೟(೟శభ)
(ଵା௥)೟
+ 60 ∑ସ௧ୀ଴
௣಴೟ ௒಴೟ ି஼಴೟
(ଵା௥)೟
(19)
The present value of the net operating revenue generated by this farm in its 6th 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,
"#஺ (6,6) = 40 ∑ଶଽ
௧ୀଶହ
௣ಲ೟ ௒ಲ೟(೟షమర) ି஼ಲ೟(೟షమర)
(ଵା௥)೟
+ 60 ∑ଶଽ
௧ୀଶହ
௣಴೟ ௒಴೟ ି஼಴೟
(ଵା௥)೟
(20)
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 6th consecutive 5 year increment or block
of almond farming is given by s = 1,…,5.
Now consider an almond orchard established on the farm at t = 10 (in the previous 5 year
increment the farm was in another land use). The present value of the net operating revenue
generated by such an operation over its first 5 years of operation (t = 10,…,14 and s = 1,…,5) is
given in element (3,1) of NRA,
"#஺ (3,1) = 40 ∑ଵସ
௧ୀଵ଴
௣ಲ೟ ௒ಲ೟(೟షవ) ି஼ಲ೟(೟షవ)
(ଵା௥)೟
+ 60 ∑ଵସ
௧ୀଵ଴
௣಴೟ ௒಴೟ ି஼಴೟
(ଵା௥)೟
(21)
The present value of the net operating revenue generated by this farm over the years 25,…,29
is given in element (6,4) of NRA
"#஺ (6,4) = 40 ∑ଶଽ
௧ୀଶହ
௣ಲ೟ ௒ಲ೟(೟షవ) ି஼ಲ೟(೟షవ)
(ଵା௥)೟
+ 60 ∑ଶଽ
௧ୀଶହ
௣಴೟ ௒಴೟ ି஼಴೟
(ଵା௥)೟
(22)
Recall that the 4th column means the 16th through 20th consecutive years of an almond farm
and therefore s = 16,…,20 for at matrix position(6,4).
The process of creating "# ௐ is identical to the process of creating "#஺ (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 "# ௉ is very similar except a new rotation begins after the
13
30th, 60th, and 95th consecutive years of a prune orchard and there are no wheat returns to
include. Finally, when creating "# ு , the net operating revenue matrix for pasture, a new
stand has to be established after 20, 40, 60, and 80 consecutive years in pasture.
9. MATLAB Code for orchard and pasture use
The variance-covariance matrix in Table 3 was created with the MATLAB function file
loglikelihoodfnwithwheat.m
The NR matrices for each crop are produced by the MATLAB function files Almonds.m,
Walnuts.m, Prunes.m, Pasture.m, and Wheat.m
The sunk and tree clearing costs that are eventually added to the net returns are determined in
the MATLAB function files AlmondSunkCost.m, AlmondClearCosts.m, WalnutSunkCost.m,
WalnutClearCost.m, PruneSunkCost.m, PastureSunkCost.m, and WheatSunkCost.m
The orchards and pasture portion of our model works thusly:
• In MainModel.m the initial price vector p and the price variance-covariance matrix are
passed into the function file Prices.m.
• Prices.m returns a 100 year price matrix to MainModel.m, one price for each unique
commodity and time period combination (a 5 x 100 matrix).
• Assumed crop input levels and input prices (defined in MainModel.m) and the price
matrix are passed into the various crop function files.
• The crop function files return the NR matrices.
• Then the function files for sunk and tree clearing costs are run (these costs are assigned
to a farm when the model evaluates all potential land use trajectories on the farm; see
below)
Every time we run the model we generate a new 100 year price matrix.
10. 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.
14
Let the per acre cost of starting up the carbon project and completing the listing be given by Lt.
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 Plt. Let the
one time per acre payment to the farmer for accepting the easement in year t be given by Et.
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 Salt, 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 BCt, BCt+1, BCt+2, BCt+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 15th through 20th years of
rotation (s = 15 – 20) then BCt would indicate the amount of woody biomass carbon stored in
on the farm at the end of the orchard’s 21st year, BCt+1 would indicate the amount of woody
biomass carbon stored on the farm at the end of the orchard’s 22nd year, etc. The series BCt,
BCt+1, BCt+2, BCt+3,… would continue to assume an almond orchard is re-established on the farm
at the end of orchard’s 25th 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 Ct, Ct+1, Ct+2, Ct+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 25th 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 BCt+1 – BCt = 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 4th, 10th, 16th, or 22nd year of the forest). After initial verification the project is
eligible to receive Offset credits as long as the afforestation project meets program
requirements.
20
Includes Inventory Cost ($30/ac); Forest Analysis ($45/ac); Verification Cost ($10/ac); and Legal Expenses
($101/ac) (or cells D54, D57, D59, and D60 of the 3_FINANCE_ASSUMPTIONS sheet in the pro forma tool).
15
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 Ct, Ct+1, Ct+2, Ct+3, …, Ct+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 Ct, Ct+1, Ct+2, Ct+3, …, Ct+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, IVt+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 SVt+v+18 and TVt+v+12. From this sampling the amount of carbon stored
on the farm is determined and the modeled series Ct, Ct+1, Ct+2, Ct+3, …, Ct+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.
Table 4: Verification timing and cost
Verification
Cost /
Initial verification
period (v)
acre
IVt+v
4
153.25
IVt+v
10
273.25
IVt+v
16
393.25
IVt+v
22
513.25
Subsequent
verification
TVt+v+12
SVt+v+18
SVt+v+24
Etc.
Verification
period
12
6
6
Cost /
acre
313.25
193.25
193.25
Notes: Costs are based on cell D76 of the 3_FINANCE_ASSUMPTIONS sheet.
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
16
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 Ct, Ct+1, Ct+2,
Ct+3, …, Ct+v trajectory used in the model. The modeler chooses which trajectory the model
uses.
Table 5: Mg of C per hectare by forest age, forest type, and carbon pool
Stand
age
0
5
15
25
35
45
55
65
75
85
95
105
115
125
live
tree
0.0
2.6
5.7
8.8
30.6
65.1
98.3
124.0
145.3
162.7
177.1
189.0
198.8
207.0
Western Oak
standing
dead
understory
tree
0.0
0.0
0.2
4.6
0.6
4.5
0.9
4.4
3.1
4.2
4.5
4.1
5.4
4.0
6.0
4.0
6.5
4.0
6.8
4.0
7.1
4.0
7.3
3.9
7.4
3.9
7.6
3.9
Mixed Conifer
total
live
tree
0.0
7.4
10.8
14.1
37.9
73.7
107.7
134.0
155.8
173.5
188.2
200.2
210.1
218.5
0.0
4.2
8.1
14.6
22.3
32.9
46.5
62.8
81.4
102.0
124.2
147.5
171.8
196.6
Mixed Riparian
standing
understory
dead tree
0.0
0.3
0.8
1.5
2.2
3.3
4.7
6.3
8.1
10.2
12.4
14.8
17.2
19.7
0.0
4.8
4.8
6.9
4.9
3.6
2.8
2.2
1.8
1.5
1.3
1.1
1.0
1.0
total
0.0
9.3
13.7
23.0
29.4
39.8
54.0
71.3
91.3
113.7
137.9
163.4
190.0
217.3
live tree
standing
dead tree
0
18.6
39.4
59.3
78.3
96.4
113.7
130
145.5
160.1
173.8
186.6
198.5
209.6
0
0.2
0.8
1.4
1.9
2.5
3.1
3.7
4.3
4.9
5.4
6
6.6
7.2
understory
0
13.1
6.9
0.8
0
0
0
0
0
0
0
0
0
0
Total
0.0
31.9
47.1
61.5
80.2
98.9
116.8
133.7
149.8
165.0
179.2
192.6
205.1
216.8
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,
%௧ା௭ = 1 − & 1 − '௧ା௭ ௧ା௭ − 1 − '௧ା௭ିଵ ௧ା௭ିଵ ! − (௧ା௭ − (௧ା௭ିଵ !
−)௧ା௭ + min{0, %௧ା௭ିଵ }
(23)
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., &ϵ[0,1) is the assumed “leakage risk” associated
with the reforestation project, C and BC are defined above, '௧ିଵ , ௧ିଵ , (௧ିଵ, and %௧ିଵ are
all equal to 0, MC is the carbon emitted during the establishment of the forest, and )௧ା௭ = 0
for all z ≥ 1. By default we assume & = 0.24.
Therefore, %௧ is given by,
%௧ = 1 − &*1 − '௧ ௧ − (௧ + − )௧
(24)
at the end of the forest’s first year (z = 0) and by
%௧ା௭ = 1 − & 1 − '௧ା௭ ௧ା௭ − 1 − '௧ା௭ିଵ ௧ା௭ିଵ ! − (௧ା௭ − (௧ା௭ିଵ !
17
+min{0, %௧ା௭ିଵ }
(25)
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, QRt+z – QRt+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. QR50+0 – 0 gives additional carbon sequestered from January 1 to
December 31 in t = 50. Further, QR50+1 – QR50+0 gives additional carbon sequestered from
January 1 to December 31 in t = 51. Further, QR50+2 – QR50+1 gives additional carbon
sequestered from January 1 to December 31 of year 52, etc.
According to the Protocol MCt = 0.202 x (12/44) x 100 if the farm was in pasture in year t – 1
and MCt = [ρ(0.429) + (1 – ρ)(0.202)] x (12/44) x 100 if the farm was in almonds, walnuts, or
prunes in year t – 1 where ρ indicates the share of the farm in the orchard and 1 – ρ the share
of the farm in wheat. The ratio (12/44) converts Mg of CO2-e per acre, given by the values
0.202 and 0.429, to Mg of C per acre. The 100 at the end of the MCt formula scales MCt up to
farm conversion emissions.
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,
௧ା௭ = %௧ା௭ *1 − 1 − ௧ା௭ +
(26)
where ௧ା௭ is the cumulative risk factor of partial or full forest destruction in year t + z.
௧ା௭ is given by,
௧ା௭ = 1 − ,௧ା௭ 1 − -௧ା௭ 1 − .௧ା௭ 1 − /௧ା௭ ×
1 − 0௧ା௭ 1 − 1௧ା௭ 1 − 2௧ା௭ 1 − ௧ା௭ (27)
where ,௧ା௭ is the financial failure risk in year t + z, -௧ା௭ is the illegal forest biomass removal risk
in year t + z, .௧ା௭ is the land-use conversion risk in year t + z, /௧ା௭ is the overharvesting risk in
year t + z, 0௧ା௭ is the social risk in year t + z, 1௧ା௭ is the wildfire risk in year t + z, 2௧ା௭ 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 ௧ା௭ = 1 and
௧ା௭ = %௧ା௭ 1 − 0 = %௧ା௭ . In other words, the famer is paid for all adjusted “additional”
sequestration created at the end of verification year t + s. If ௧ା௭ < 1 then ௧ା௭ < %௧ା௭
and the farmer is obliged to place %௧ା௭ − 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 1௧ା௭ = 0.04 for all t + z. Because we assume lands are covered by a qualified
conservation easement, the default risk for ,௧ା௭ is 0.01 for all t + z, the default risk for .௧ା௭ is 0
18
for all t + z, and the default risk for /௧ା௭ is 0 for all t + z. The other risk factors are set by
default for any U.S. forest project at these values: -௧ା௭ is 0 for all t + z, 0௧ା௭ is 0.02 for all t + z,
2௧ା௦ is 0.03 for all t + z, and ௧ା௭ is 0.03 for all t + z. Therefore, equation (27) in our model is
equal to,
௧ା௭ = 0.991110.980.960.970.97 = 0.876
(28)
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 NRf.
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)) and the fourth dimension indicates if the previous land use
would have been entering its establishment year (s = 1), its 6th year (s = 6), 11th year (s = 11),
16th year (s = 16), 21st year (s = 21), or its 26th year (s = 26). (Recall the model only allows
considers land use choice every 5 years.) Only prune orchards can reach a 26th year of rotation.
Finally, the rotation year of a pasture is irrelevant when it comes to baseline carbon. Therefore,
the 4th dimension is undefined for pasture
To illustrate "#௙ = #௙ − 3௙ consider a forest established at the beginning of some arbitrary 5year increment or block that begins in year t (let us index this block with x). Assume the farm
would have otherwise re-established an almond orchard during the 5-year block. Assume the
initial verification occurs in the fourth year (v = 3) of the forest. The present value of the net
revenue generated by such an operation over its first 5 years is given at matrix element x,1,1,1
and is equal to,
#௙ x, 1,1,1 =
ௌ௔௟೟ ାଵ଴଴ா೟
ሺଵା௥ሻ೟
+
(௣೟శయ ି௧௥೟శయ )஼௉೟శయ (:,:,ଵ,ଵ)
ሺଵା௥ሻ೟శయ
(29)
where ௧ is the real price of a carbon permit (an adjusted metric ton of carbon) at the end of
year t, trt = 0.48 is the transition cost per permit (or metric ton) in year t, and ௧ା௩ (: , : ,1,1)
gives the carbon credits issued in verification year v. 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 3௙ (1,1,1,1) is given by,
೟
೟
೟శయ
+ ሺଵା௥ሻ
+ ∑ସ௭ୀ଴ ሺଵା௥ሻ೟శ೥
5
3ࢌ x, 1,1,1 = 100 4ሺଵା௥ሻ
೟
೟శయ
೟శ೥
௅ ା௉௟
ூ௏
஺ோ
(30)
where the cost of an annual report, given by ARt, is $20 per acre.21
21
Annual reports must be submitted to Protocol authorities (see section 9.2.1 of the Protocol). Includes Annual
Reporting Costs ($1500) and Annual Membership Fee ($500) (or cells D79 and D80 of the
3_FINANCE_ASSUMPTIONS sheet in the pro forma tool). (Saah et al. 2014)
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Now consider this same operation in the next two 5-year increments,
#௙ (x + 1,2,1,1) = 0
(31)
#௙ (x + 2,3,1,1) = 0
(32)
and
3ࢌ x + 1,2,1,1 = 100 4∑ଽ௭ୀହ ሺଵା௥ሻ೟శ೥
5
೟శ೥
(33)
೟శ೥
3ࢌ x + 2,3,1,1 = 100 4∑ଵସ
௭ୀଵ଴ ሺଵା௥ሻ೟శ೥ 5
(34)
஺ோ
஺ோ
5-year revenues in (31) and (32) are zero because no verification occurs in those years. In the
fourth 5-year increment we have the t + v + 12 – year verification:
#௙ x + 3,4,1,1 =
(௣೟శభఱ ି௧௥೟శభఱ )஼௉೟శభఱ (:,:,ଵ,ଵ)
ሺଵା௥ሻ೟శభఱ
(35)
and
೟శభఱ
೟శ೥
∑ଵଽ
3ࢌ x + 3,4,1,1 = 100 4ሺଵା௥ሻ
௭ୀଵହ ሺଵା௥ሻ೟శ೥ 5
೟శభఱ +
்௏
஺ோ
(36)
In the fifth 5-year increment we have the t + v + 18 – year verification:
#௙ x + 4,5,1,1 =
(௣೟శమభ ି௧௥೟శమభ )஼௉೟శమభ (:,:,ଵ,ଵ)
ሺଵା௥ሻ೟శమభ
(37)
and
೟శమభ
೟శ೥
∑ଶସ
3ࢌ x + 4,5,1,1 = 100 4ሺଵା௥ሻ
௭ୀଶ଴ ሺଵା௥ሻ೟శ೥ 5
೟శమభ +
ௌ௏
஺ோ
(38)
Now assume the baseline land use was a prune orchard that would have been entering its 16th
year at t. In this case, we would use #௙ x, 1,3,4 and 3௙ x, 1,3,4 for the first block of 5-years,
etc.
12. MATLAB Code for forest use
The model works thusly:
• In MainModel.m the carbon credit price vector, the forest type selected, and other
inputs needed to determine the net returns to forest use are defined.
• All of these inputs are passed into the function file CarbonCreated.m
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•
o CarbonCreated.m uses the MATLAB functions SmithCalculator.m, QR.m,
Credits.m, ValueofCredits.m, and CostofOffset.m.
CarbonCreated.m passes back 4 matrices of net returns to forest use.
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 6th year when the forest was established. The third block of 20 rows
assumes the baseline almond orchard would have begun its 16th year when the
forest was established. The fourth block of 20 rows assumes the baseline almond
orchard would have begun its 20th 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 25th 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.
13. 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 5th year and the conversion to forest is
irreversible. For example, the land use vector,
[1 1 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4]
(39)
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]
(40)
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]
(41)
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
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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
11th year of rotation in the year the forest was established.
Table 6: Forest land use codes
Previous land use Previous rotation block
1
0, 5, 10, 15, 20
1
6, 11, 16
1
7, 12, 17
1
8, 13, 18
1
9, 14, 19
2
5, 10, 15, 20
2
6, 11, 16
2
7, 12, 17
2
8, 13, 18
2
9, 14, 19
3
6, 12, 18
3
1, 7, 13, 19
3
2, 8, 14, 20
3
3, 9, 15
3
4, 10, 16
3
5, 11, 17
4
Any
Land code
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
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]
(42)
[1 2 3 1 2 3 4 5 6 7 1 2 3 1 2 3 4 5 6 7 8 9 10]
(43)
[1 2 3 1 2 3 4 5 6 7 1 2 3 1 2 3 4 5 6 7 8 9 10]
(44)
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,
22
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.
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