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

Forest and Agricultural Sector Optimization Model (FASOM)

Basic Mathematical Structure

Linear Programming

j jj

ij j ij

j

c X

. . a X b for all i

X 0 for all j

Max

s t

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

Important Equations

Objective Function

Resource Restrictions

Commodity Restrictions

Intertemporal Transition Restrictions

Emission Restrictions

Parameter Description

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

Variable Unit Type DescriptionCROP 1E3 ha 0 Crop productionPAST 1E3 ha 0 Pasture LIVE mixed 0 Livestock raisingFEED mixed 0 Animal feeding TREE 1E3 ha 0 Standing forestsHARV 1E3 ha 0 harvestingBIOM 1E3 ha 0 Biomass crop plantations for bioenergy ECOL 1E3 ha 0 Wetland ecosystem reservesLUCH 1E3 ha 0 Land use changesRESR mixed 0 Factor and resource usagePROC mixed 0 Processing activitiesSUPP 1E3 t 0 SupplyDEMD 1E3 t 0 DemandTRAD 1E3 t 0 TradeEMIT mixed Free Net emissionsSTCK mixed 0 Environmental and product stocksWELF 1E6 € Free Economic SurplusCMIX - 0 Crop Mix

Index Symbol ElementsTime Periods t 2005-2010, 2010-2015, …, 2145-2150Regions r 25 EU member states, 11 Non-EU international regions Species s All individual and aggregate species categories

Crops c(s) Soft wheat, hard wheat, barley, oats, rye, rice, corn, soybeans, sugar beet, potatoes, rapeseed, sunflower, cotton, flax, hemp, pulse

Trees f(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

Perennials b(s) Miscanthus, Switchgrass, Reed Canary Grass, Poplar, , Arundo, Cardoon, Eucalyptus Livestock l(s) Dairy, beef cattle, hogs, goats, sheep, poultry Wildlife w(s) 43 Birds, 9 mammals, 16 amphibians, 4 reptilesProducts y 17 crop, 8 forest industry, 5 bioenergy, 10 livestockResources/Inputs i Soil types, hired and family labor, gasoline, diesel, electricity, natural gas, water, nutrients Soil types j(i) Sand, loam, clay, bog, fen, 7 slope, 4 soil depth classes Nutrients n(i) Dry matter, protein, fat, fiber, metabolic energy, Lysine

Technologies m alternative tillage, irrigation, fertilization, thinning, animal housing and manure management choices

Site quality q Age and suitability differences Ecosystem state x(q) Existing, suitable, marginal Age cohorts a(q) 0-5, 5-10, …, 295-300 [years]Soil state v Soil organic classesStructures u FADN classifications (European Commission 2008) Size classes z(u) < 4, 4 - < 8, 8 - < 16, 16 - < 40, 40- < 100, >= 100 all in ESU (European Commission 2008)

Farm specialty o(u) Field crops, horticulture, wine yards, permanent crops, dairy farms, grazing livestock, pigs and or poultry, mixed farms

Altitude levels h(u) < 300, 300 – 600, 600 – 1100, > 1100 meters

Environment e 16 Greenhouse gas accounts, wind and water erosion, 6 nutrient emissions, 5 wetland types

Policies p Alternative 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*

ProducerSurplus

ConsumerSurplus

t

t

TREEt t r, j,v,f ,u,a ,m,p r,T, j,v,f ,u,a ,m,p

t t r, j,v,f ,u,a ,m,p

tt

CS

Max WELF RS TREE

C

Basic Objective Function

Terminal value of standing forests

Discount factor Consumer surplusResource surplusCosts of production and trade

r,t ,yr,y

t t r,t ,yr,y

r,t ,ir,i

DEMD d

CS RS SUPP d

RESR d

DEMDr,t,y

SUPPr,t,y

RESRr,t,i

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

0 0, p ,qDEMDr,t,y

• Decreasing marginal revenues• uniquely defined by

• constant elasticity function• observed price-quantity pair (p0,q0) • estimated elasticity (curvature)

price

sales

Demand function

q00

p0

q0

q(p) p

p q

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

CROPr,t , j,v,c,u,q,m,p r,t , j,v,c,u,q,m,p

r, j,v,c,u,q,m,p

PASTr,t , j,v,s,u,q,m,p r,t , j,v,s,u,q,m,p

r, j,v,s,u,q,m,p

BIOMr,t , j,v,b,u,q,m,p r,t , j,v,b,u,q,m,p

r, j,v,b,u,q,m,p

r,t , j,v,f ,u,a

t

CROP

PAST

BIOM

C

HARV,m,p r,t , j,v,f ,u,a ,m,p

r, j,v,f ,u,a ,m,p

TREEr,t , j,v,f ,u,a ,m,p r,t , j,v,f ,u,a ,m,p

r, j,v,f ,u,a ,m,p

ECOLr,t , j,v,s,u,x,m,p r,t , j,v,s,u,x,m,p

r, j,v,s,u,x,m,p

LIVEr,t ,l,u,m,p r,t ,l,u,m,

HARV

TREE

ECOL

LIVE

pr,l,u,m,p

PROC FEEDr,t ,m r,t ,m r,t ,l,m r,t ,l,m

r,m r,l,m

LUCHr,t , j,s,u,u r,t , j,s,u,u

r, j,u,u

TRADEr,r,t ,y r,r ,t ,y

r,r ,y

PROC FEED

LUCH

TRAD

Production and Trade

Cost

ResourceAccounting Equations

(r,t,i)

CROPr,t , j,v,c,u,q,m,p,i r,t , j,v,c,u,q,m,p

j,v,c,u,q,m,p

PASTr,t , j,v,s,u,q,m,p,i r,t , j,v,s,u,q,m,p

j,v,s,u,q,m,p

BIOMr,t , j,v,b,u,q,m,p,i r,t , j,v,b,u,q,m,p

j,v,b,u,q,m,p

r,t , j,v,f ,u,a ,m

CROP

PAST

BIOM

HARV,p,i r,t , j,v,f ,u,a ,m,p

j,v,f ,u,a ,m,p

TREEr,t , j,v,f ,u,a ,m,p,i r,t , j,v,f ,u,a ,m,p

j,v,f ,u,a ,m,p

ECOLr,t , j,v,s,u,x,m,p,i r,t , j,v,s,u,x,m,p

j,v,s,u,x,m,p

LIVEr,t ,l,u,m,p,i r,t ,l,u,m,

HARV

TREE

ECOL

LIVE

r,t ,i

pl,u,m,p

PROCr,t ,m,i r,t ,m

m

FEEDr,t ,l,m,i r,t ,l,m

m

RESR

PROC

FEED

Physical Resource Limits(r,t,i)

r,t ,i r,t ,iRESR

CROPr,t , j,v,c,u,q,m,p,y r,t , j,v,c,u,q,m,p

j,v,c,u,q,m,p

PASTr,t , j,v,s,u,q,m,p,y r,t , j,v,

PROCr,t ,m,y r,t ,m

m

FEEDr,t ,l,m,y r,t ,l,m

m

r,r ,t ,yr

r,t ,y

CROP

PAST

PROC

FEED

TRAD

DEMD

s,u,q,m,pj,v,s,u,q,m,p

BIOMr,t , j,v,b,u,q,m,p,y r,t , j,v,b,u,q,m,p

j,v,b,u,q,m,p

HARVr,t , j,v,f ,u,a ,m,p,y r,t , j,v,f ,u,a ,m,p

j,v,f ,u,a ,m,p

TREEr,t , j,v,f ,u,a ,m,p,y r,t , j,v,f ,u,a ,m,p

j,v,f ,u,

BIOM

HARV

TREE

a,m,p

ECOLr,t , j,v,s,u,x,m,p,y r,t , j,v,s,u,x,m,p

j,v,s,u,x,m,p

LIVEr,t ,l,u,m,p,y r,t ,l,u,m,p

l,u,m,p

r,r,t ,yr

r,t ,y

ECOL

LIVE

TRAD

SUPP

Commodity Equations (r,t,y)

Demand Supply

PROCr,t ,m,y r,t ,m

m

PROC 0

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

r,t 1, j,v,f ,u,a 1,m,p t 1 a 1r,t , j,v,f ,u,a ,m,p a 1

r,t 1, j,v,f ,u,a ,m,p t 1 a Ar,t , j,v,f ,u,a ,m,p a 1

r, j,v,f ,u,a ,m,p t 1

TREETREE

TREEHARV

INIT

Emission Accounting

Equation(r,t,e)

CROPr,t,j,v,c,u,q,m,p,e r,t , j,v,c,u,q,m,p

j,v,c,u,q,m,p

PASTr,t,j,v,c,u,q,m,p,e r,t , j,v,c,u,q,m,p

j,v,c,u,q,m,p

BIOMr,t,j,v,b,u,q,m,p,e r,t , j,v,b,u,q,m,p

j,v,b,u,q,m,p

r,t,j

r,t ,e

CROP

PAST

BIOM

EMIT

TREE,v,f,u,a,m,p,e r,t , j,v,f ,u,a ,m,p

j,v,f ,u,a,m,p

ECOLr,t,j,v,s,u,x,m,p,e r,t , j,v,s,u,x,m,p

j,v,s,u,x,m,p

LIVEr,t,s,u,m,p,e r,t,s,u,m,p

s,u,m,p

LUCHr,t ,s,u, , ,e r,t ,s,u, ,

s,u, ,

TREE

ECOL

LIVE

LUCH

PROCr,t ,m,e r,t ,m

m

FEEDr,t ,l,m,e r,t ,l,m

m,l

STCKr,t ,d,e r,t ,d r,t 1,d

d

PROC

FEED

STCK STCK

Environmental Policy

r,t ,e r,t ,eEMIT

r,t ,e r,t ,er,t ,e

WELF ( ) EMIT or

Duality restrictions (r,t,u)

• Prevent extreme specialization• Incorporate difficult to observe data• Calibrate model based on duality theory• May include „flexibility contraints“

CMIXr,t , j,v,c,u,q,m,p r,t ,c,u r,t ,t ,u

j,v,c,q,m,p t

CROP CMIX

Past periods

Observed crop mixes

Crop Mix VariableNo crop (c) index!

Crop Area Variable

Miscellaneous

• GAMS• Systematic Model Check• Linearization• Alternative Objective Function

Linear Program Duality

i allfor 0 U

j allfor caUs.t.

ZbUMin

i

ji

iji

iii

jallfor0X

iallforbXa..

ZXc

j

ij

jij

jjj

ts

Max

Reduced Cost

j

ji

iij X

ZcUa

Shadow prices

TechnicalCoefficients

ObjectiveFunctionCoefficients

Variable Decomposition Example (not from FASOM)

## Landuse_Var(Bavaria,Sugarbeet) SOLUTION VALUE 1234.00

EQN Aij Ui Aij*Uiobjfunc_Equ 350.00 1.0000 350.00Endowment_Equ(Bavaria,Land) 1.0000 90.000 90.000Endowment_Equ(Bavaria,Water) 250.00 0.0000 0.0000Production_Equ(Bavaria,Sugarbeet) -11.000 40.000 -440.00TRUE REDUCED COST 0.0000

j

ji

iij X

ZcUa

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

j

ji

iij X

ZcUa

Complementary Slackness

*' *

*' *

0U b AX

0U A C X

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