# Forest and Agricultural Sector Optimization Model (FASOM) Basic Mathematical Structure.

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Forest and Agricultural Sector Optimization Model (FASOM) Basic Mathematical Structure

Linear Programming FASOM can solve up to 6 Million Variables (j), 1 Million Equations (i)

Important Equations Objective Function Objective Function Resource Restrictions Resource Restrictions Commodity Restrictions Commodity Restrictions Intertemporal Transition Restrictions Intertemporal Transition Restrictions Emission Restrictions Emission Restrictions

ParameterDescription  Technical coefficients (yields, requirements, emissions)  Objective function coefficients  Supply and demand functions  Supply and demand function elasticities  Discount rate, product depreciation, dead wood decomposition  Resource endowments Soil state transition probabilities  Land use change limits  Initial or previous land allocation  Alternative objective function parameters

VariableUnitTypeDescription CROP1E3 ha  0 Crop production PAST1E3 ha  0 Pasture LIVEmixed  0 Livestock raising FEEDmixed  0 Animal feeding TREE1E3 ha  0 Standing forests HARV1E3 ha  0 harvesting BIOM1E3 ha  0 Biomass crop plantations for bioenergy ECOL1E3 ha  0 Wetland ecosystem reserves LUCH1E3 ha  0 Land use changes RESRmixed  0 Factor and resource usage PROCmixed  0 Processing activities SUPP1E3 t  0 Supply DEMD1E3 t  0 Demand TRAD1E3 t  0 Trade EMITmixedFreeNet emissions STCKmixed  0 Environmental and product stocks WELF1E6 €FreeEconomic Surplus CMIX-  0 Crop Mix

IndexSymbolElements Time Periodst2005-2010, 2010-2015, …, 2145-2150 Regionsr25 EU member states, 11 Non-EU international regions SpeciessAll individual and aggregate species categories Cropsc(s) Soft wheat, hard wheat, barley, oats, rye, rice, corn, soybeans, sugar beet, potatoes, rapeseed, sunflower, cotton, flax, hemp, pulse Treesf(s) Spruce, larch, douglas fir, fir, scottish pine, pinus pinaster, poplar, oak, beech, birch, maple, hornbeam, alnus, ash, chestnut, cedar, eucalyptus, ilex locust, 4 mixed forest types Perennialsb(s)Miscanthus, Switchgrass, Reed Canary Grass, Poplar,, Arundo, Cardoon, Eucalyptus Livestockl(s)Dairy, beef cattle, hogs, goats, sheep, poultry Wildlifew(s)43 Birds, 9 mammals, 16 amphibians, 4 reptiles Productsy17 crop, 8 forest industry, 5 bioenergy, 10 livestock Resources/InputsiSoil types, hired and family labor, gasoline, diesel, electricity, natural gas, water, nutrients Soil typesj(i)Sand, loam, clay, bog, fen, 7 slope, 4 soil depth classes Nutrientsn(i)Dry matter, protein, fat, fiber, metabolic energy, Lysine Technologiesm alternative tillage, irrigation, fertilization, thinning, animal housing and manure management choices Site qualityqAge and suitability differences Ecosystem statex(q)Existing, suitable, marginal Age cohortsa(q)0-5, 5-10, …, 295-300 [years] Soil statevSoil organic classes StructuresuFADN classifications (European Commission 2008) Size classesz(u) = 100 all in ESU (European Commission 2008) Farm specialtyo(u) Field crops, horticulture, wine yards, permanent crops, dairy farms, grazing livestock, pigs and or poultry, mixed farms Altitude levelsh(u) 1100 meters Environmente16 Greenhouse gas accounts, wind and water erosion, 6 nutrient emissions, 5 wetland types PoliciespAlternative policies

Objective Function Maximize +Area underneath demand curves -Area underneath supply curves -Costs ±Subsidies / Taxes from policies The maximum equilibrates markets!

Market Equilibrium Demand Supply Price Quantity P* Q* Producer Surplus Consumer Surplus

Basic Objective Function Terminal value of standing forests Discount factorConsumer surplus Resource surplus Costs of production and trade

Consumer and Resource Surplus

Economic Principles Rationality ("wanting more rather than less of a good or service") Law of diminishing marginal returns Law of increasing marginal cost

Demand function Area underneath demand function Decreasing marginal revenues uniquely defined by constant elasticity function observed price-quantity pair (p 0,q 0 ) estimated elasticity  (curvature) price sales Demand function q 00 p0p0 q0q0

Economic Surplus Maximization Implicit Supply and Demand Forest InventoryLand Supply Water Supply Labor Supply Animal Supply National Inputs Import Supply Processing Demand Feed Demand Domestic Demand Export Demand CS PS

Resource Accounting Equations (r,t,i)

Physical Resource Limits (r,t,i)

Commodity Equations (r,t,y) Demand  Supply

Industrial Processing (r,t,y) Processing activities can be bounded (capacity limits) or enforced (e.g. when FASOM is linked to other models)

Forest Transistion Equations Standing forest area today + harvested area today <= forest area from previous period Equation indexed by r,t,j,v,f,u,a,m,p

Emission Accounting Equation (r,t,e)

Environmental Policy or

Duality restrictions (r,t,u) Prevent extreme specialization Incorporate difficult to observe data Calibrate model based on duality theory May include „flexibility contraints“ Past periods Observed crop mixes Crop Mix Variable No crop (c) index! Crop Area Variable

Miscellaneous GAMS Systematic Model Check Linearization Alternative Objective Function

Linear Program Duality

Reduced Cost Shadow prices Technical Coefficients Objective Function Coefficients

Variable Decomposition Example (not from FASOM) ## Landuse_Var(Bavaria,Sugarbeet) SOLUTION VALUE 1234.00 EQN Aij Ui Aij*Ui objfunc_Equ 350.00 1.0000 350.00 Endowment_Equ(Bavaria,Land) 1.0000 90.000 90.000 Endowment_Equ(Bavaria,Water) 250.00 0.0000 0.0000 Production_Equ(Bavaria,Sugarbeet) -11.000 40.000 -440.00 TRUE REDUCED COST 0.0000

Variable Decomposition Example (not from FASOM) ## Landuse_Var(Bavaria,Wheat) SOLUTION VALUE 0.00000 EQN Aij Ui Aij*Ui objfunc_Equ 350.00 1.0000 350.00 Endowment_Equ(Bavaria,Land) 1.0000 250.00 250.00 Endowment_Equ(Bavaria,Water) 250.00 0.0000 0.0000 Production_Equ(Bavaria,Wheat) -1.0000 89.000 -89.000 TRUE REDUCED COST 511.00

Complementary Slackness Reduced Cost Opt. Variable Level Shadow Price Opt. Slack Variable Level

Solution Decomposition Insights Why is an activity not used? How do individual equations contribute to the variable‘s optimality?

Current work Land management adaptation to policy & development Externality mitigation (Water, Greenhouse Gases, Biodiversity, Soil fertility) Stochastic formulation (extreme events) Land use & management change costs Learning and agricultural research policies Investment restrictions

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