Calculation of holding income

MtM ABM description

Baseline income equation for holdings

The baseline profit Z of each hectare of a holding is given by

Z=Y-X

where Y is gross income, and X is total costs. These are in turn given by

\begin{array}{rcl} Y &=& pq \\ X &=& x+te \end{array}

where

• p is the price obtained per unit,
• q is the quantity produced per hectare,
• x is the cost per hectare to operate,
• e is the greenhouse gas emissions per hectare, and
• t is the tax on each unit of emissions.

Each of these parameters may depend on the farm type F and the land use capability (LUC), L, such that the full calculation of profit for each hectare of a holding is Z_{FL} = p_Fq_{FL} - x_{FL} - te_{FL} and total holding profit is obtaining by summing this quantity across all hectares of the holding. Overall profit of a farm is determined by summation across all its constituent holdings. Note that while currently all cells in a holding have the same LUC this is not a requirement and the calculation will work equally well if LUC were to vary across a holding.

Parameter values

The parameter values are stored in a number of CSV files which are read in by the model at start up from the .data/market/<scenario> folder specified.

Prices

p_F and the greenhouse gas tax (i.e. carbon price), t, are in the prices.csv file:

prices <- read.csv(str_glue("{base_dir}prices.csv"), 
                   header = T, row.names = 1) |> 
  slice(1:2) # only rows 1 & 2 relevant
prices |> 
  kable() |> 
  kable_styling("striped", full_width = F)

	SNB	Dairy	Forest	Crop
Price_Commodity	5	7.5	157	0.5
Price_GhG	25	25.0	25	25.0

Yields

Yields by farm type and land use capability, q_{FL}, are in the commodity-yields.csv file:

yields <- read.csv(str_glue("{base_dir}commodity-yields.csv"), 
                   header = T, row.names = 1)
yields |> 
  kable() |> 
  kable_styling("striped", full_width = F) |>
  scroll_box(height = "250px")

	SNB	Dairy	Forest	Crop
LUC1_Mean	768.505660	1503.0000	30.0	9667.000
LUC1_SD	153.701132	300.6000	6.0	1933.400
LUC2_Mean	581.247332	1283.0000	29.0	9183.650
LUC2_SD	116.249466	256.6000	5.8	1836.730
LUC3_Mean	372.371287	1174.3000	28.0	8724.468
LUC3_SD	74.474257	234.8600	5.6	1744.893
LUC4_Mean	332.178378	923.3900	27.0	8288.244
LUC4_SD	66.435676	184.6780	5.4	1657.649
LUC5_Mean	259.480336	877.7308	26.0	7873.832
LUC5_SD	51.896067	175.5462	5.2	1574.766
LUC6_Mean	207.386971	845.7812	25.0	7480.140
LUC6_SD	41.477394	169.1562	5.0	1496.028
LUC7_Mean	193.188428	700.0000	24.0	7106.133
LUC7_SD	38.637686	140.0000	4.8	1421.227
LUC8_Mean	49.945208	661.0000	23.0	6750.827
LUC8_SD	9.989042	132.2000	4.6	1350.165

Costs

Base costs per ha. by farm type and land use capability, x_{FL}, are in the input-costs.csv file:

costs <- read.csv(str_glue("{base_dir}input-costs.csv"), 
                   header = T, row.names = 1)
costs |> 
  kable() |> 
  kable_styling("striped", full_width = F) |>
  scroll_box(height = "250px")

	SNB	Dairy	Forest	Crop
LUC1_Mean	3500	9500	4000	3000
LUC1_SD	700	1900	800	600
LUC2_Mean	2650	8050	4000	2850
LUC2_SD	530	1610	800	570
LUC3_Mean	1625	7350	3850	2675
LUC3_SD	325	1470	770	535
LUC4_Mean	1450	5500	3700	2500
LUC4_SD	290	1100	740	500
LUC5_Mean	1110	5200	3550	2400
LUC5_SD	222	1040	710	480
LUC6_Mean	875	5000	3500	2200
LUC6_SD	175	1000	700	440
LUC7_Mean	810	4000	3375	2100
LUC7_SD	162	800	675	420
LUC8_Mean	125	4000	3250	2000
LUC8_SD	25	800	650	400

Emissions

Emissions per ha. by farm type and land use capability, e_{FL}, are in the ghg-emissions.csv file:

emissions <- read.csv(str_glue("{base_dir}ghg-emissions.csv"), 
                      header = T, row.names = 1)
emissions |> 
  kable() |> 
  kable_styling("striped", full_width = F) |>
  scroll_box(height = "250px")

	SNB	Dairy	Forest	Crop
LUC1_Mean	4.000	11.00	-15.000000	1.2000000
LUC1_SD	1.200	3.30	4.500000	0.3600000
LUC2_Mean	3.750	10.50	-14.250000	1.1400000
LUC2_SD	1.125	3.15	4.275000	0.3420000
LUC3_Mean	3.500	10.00	-13.537500	1.0830000
LUC3_SD	1.050	3.00	4.061250	0.3249000
LUC4_Mean	3.250	9.50	-12.860625	1.0288500
LUC4_SD	0.975	2.85	3.858188	0.3086550
LUC5_Mean	3.000	9.00	-12.217594	0.9774075
LUC5_SD	0.900	2.70	3.665278	0.2932223
LUC6_Mean	2.750	8.50	-11.606714	0.9285371
LUC6_SD	0.825	2.55	3.482014	0.2785611
LUC7_Mean	2.500	8.00	-11.026378	0.8821103
LUC7_SD	0.750	2.40	3.307914	0.2646331
LUC8_Mean	2.250	7.50	-10.475059	0.8380048
LUC8_SD	0.675	2.25	3.142518	0.2514014

 

Note that all of yields, base costs, and emissions are stored as a mean and a standard deviation, and that each holding will see these numbers vary every model time period by drawing from a normal distribution.

We can use these data to show the profit, gross income, and total costs for each farm type across different land use capability classes. We’ll throw away the standard deviations for simplicity and consider only mean outcomes.

prices <- prices |> 
  t() |> 
  as.data.frame() |> 
  tibble::rownames_to_column("farm_type") |>
  rename(price = Price_Commodity, carbon_tax = Price_GhG) |>
  slice(rep(row_number(), 8)) |>
  mutate(LUC = rep(1:8, each = 4))

yields <- yields |>
  slice(seq(1, 15, 2)) |> 
  t() |> 
  as.data.frame() |> 
  tibble::rownames_to_column("farm_type") |>
  pivot_longer(-1) |>
  mutate(LUC = rep(1:8, 4), yield = round(value, 1)) |>
  select(farm_type, LUC, yield)

costs <- costs |>
  slice(seq(1, 15, 2)) |> 
  t() |> 
  as.data.frame() |> 
  tibble::rownames_to_column("farm_type") |>
  pivot_longer(-1) |>
  mutate(LUC = rep(1:8, 4), costs = round(value, 1)) |>
  select(farm_type, LUC, costs)

emissions <- emissions |> 
  slice(seq(1, 15, 2)) |> 
  t() |> 
  as.data.frame() |> 
  tibble::rownames_to_column("farm_type") |>
  pivot_longer(-1) |>
  mutate(LUC = rep(1:8, 4), emissions = round(value, 1)) |>
  select(farm_type, LUC, emissions)

combined_data <- prices |>
  left_join(yields) |>
  left_join(costs) |>
  left_join(emissions) |>
  mutate(gross_income = price * yield, # LUC = as.factor(LUC),
         total_costs = costs + carbon_tax * emissions, base_costs = costs,
         profit = gross_income - total_costs) |>
  select(farm_type, LUC, profit, price, yield, gross_income, base_costs,
         emissions, carbon_tax, total_costs) |>
  rename(Profit = profit, `Gross income` = gross_income, `Total costs` = total_costs)

combined_data |> 
  kable() |> 
  kable_styling("striped", full_width = F) |>
  scroll_box(height = "250px")

farm_type	LUC	Profit	price	yield	Gross income	base_costs	emissions	carbon_tax	Total costs
SNB	1	242.50	5.0	768.5	3842.50	3500	4.0	25	3600.0
Dairy	1	1497.50	7.5	1503.0	11272.50	9500	11.0	25	9775.0
Forest	1	1085.00	157.0	30.0	4710.00	4000	-15.0	25	3625.0
Crop	1	1803.50	0.5	9667.0	4833.50	3000	1.2	25	3030.0
SNB	2	161.00	5.0	581.2	2906.00	2650	3.8	25	2745.0
Dairy	2	1310.00	7.5	1283.0	9622.50	8050	10.5	25	8312.5
Forest	2	908.00	157.0	29.0	4553.00	4000	-14.2	25	3645.0
Crop	2	1714.30	0.5	9183.6	4591.80	2850	1.1	25	2877.5
SNB	3	149.50	5.0	372.4	1862.00	1625	3.5	25	1712.5
Dairy	3	1207.25	7.5	1174.3	8807.25	7350	10.0	25	7600.0
Forest	3	883.50	157.0	28.0	4396.00	3850	-13.5	25	3512.5
Crop	3	1659.75	0.5	8724.5	4362.25	2675	1.1	25	2702.5
SNB	4	131.00	5.0	332.2	1661.00	1450	3.2	25	1530.0
Dairy	4	1188.00	7.5	923.4	6925.50	5500	9.5	25	5737.5
Forest	4	861.50	157.0	27.0	4239.00	3700	-12.9	25	3377.5
Crop	4	1619.10	0.5	8288.2	4144.10	2500	1.0	25	2525.0
SNB	5	112.50	5.0	259.5	1297.50	1110	3.0	25	1185.0
Dairy	5	1157.75	7.5	877.7	6582.75	5200	9.0	25	5425.0
Forest	5	837.00	157.0	26.0	4082.00	3550	-12.2	25	3245.0
Crop	5	1511.90	0.5	7873.8	3936.90	2400	1.0	25	2425.0
SNB	6	92.00	5.0	207.4	1037.00	875	2.8	25	945.0
Dairy	6	1131.00	7.5	845.8	6343.50	5000	8.5	25	5212.5
Forest	6	715.00	157.0	25.0	3925.00	3500	-11.6	25	3210.0
Crop	6	1517.55	0.5	7480.1	3740.05	2200	0.9	25	2222.5
SNB	7	93.50	5.0	193.2	966.00	810	2.5	25	872.5
Dairy	7	1050.00	7.5	700.0	5250.00	4000	8.0	25	4200.0
Forest	7	668.00	157.0	24.0	3768.00	3375	-11.0	25	3100.0
Crop	7	1430.55	0.5	7106.1	3553.05	2100	0.9	25	2122.5
SNB	8	69.50	5.0	49.9	249.50	125	2.2	25	180.0
Dairy	8	770.00	7.5	661.0	4957.50	4000	7.5	25	4187.5
Forest	8	623.50	157.0	23.0	3611.00	3250	-10.5	25	2987.5
Crop	8	1355.40	0.5	6750.8	3375.40	2000	0.8	25	2020.0

Or visually…

Below are shown the income, costs, and net revenue per hectare for each farm type across the eight land use capability classes. These are crude estimates dating back to the ARLUNZ model¹ and will be updated in due course. Even so it is expected that the relative profitability of the different broad land use classes will remain similar.

plot_data <- combined_data |> 
  pivot_longer(cols = c("Gross income", "Profit", "Total costs"))

ggplot(plot_data |> filter(name %in% c("Gross income", "Total costs"))) +
  geom_col(aes(x = as.factor(LUC), y = value, group = name, fill = name),
           position = "dodge") + 
  scale_fill_manual(values = c("#999999", "#ff6666"), name = "Income vs. Costs") +
  geom_point(data = plot_data |> filter(name == "Profit"), 
            aes(x = LUC, y = value, shape = name), size = 7) +
  scale_shape_manual(values = "\u2014", name = "") + # \u2014 is an em-dash
  # The below more correct, but doesn't extend the lines to the full width of the plot
  # geom_step(data = plot_data |> filter(name == "Profit"), 
  #           aes(x = LUC, y = value, group = farm_type, colour = name), 
  #           direction = "mid", lwd = 0.65) +
  # scale_colour_manual(values = c("black"), name = "") +
  xlab("Landuse Capability (LUC)") +
  ylab("Profit, $ per ha.") +
  facet_wrap(~ farm_type) +
  theme_minimal()

Update History

Date	Changes
2025-02-19	Split out from previous ‘core loop’ document.
2024-07-28	Correction in equation summarising full calculation of profit. Added note to consider the nudge calculation’s possible bias favouring changes after bad years.
2024-07-05	Corrected relative change from a percentage to a proportion.
2024-07-04	Added detail on calculation of relative change noting that profit can’t be handled this way because it is \pm and relative change is meaningless for such variables.
2024-07-03	Added bar chart of farm income/costs estimates.
2024-07-02	Initial post.

Footnotes

1. See: Morgan FJ and AJ Daigneault. 2015. Estimating Impacts of Climate Change Policy on Land Use: An Agent-Based Modelling Approach. PLOS ONE 10(5): e0127317.