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CAPRI CAPRI
CAPRI market model
Torbjörn Jansson*Markus Kempen
*Corresponding author+49-228-732323www.agp.uni-bonn.de
Department for Economic and Agricultural PolicyBonn UniversityNussallee 2153115 Bonn, Germany
CAPRI Training Session in WarzawJune 26-30, 2006
CAPRICommon Agricultural Policy Regional Impact
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
2
Outline
• About multi-commodity models• Principles of the CAPRI market module MultReg
step by step– Final demand– Price transmission– Production and processing
• Iterative solution• (Calibration issues)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
3
What is a Multi-Commodity Model ?
• More than one output market, but not general equilibrium
• System of equations: no objective function
• Same number of endogenous variables as equations (so called square system, CNS)
• Many examples:
– SWOPSIM (http://usda.mannlib.cornell.edu/data-sets/trade/92012/)
– AGLink OECD
– FAPRI (http://www.fapri.missouri.edu/)
– AgMemod (http://tnet.teagasc.ie/agmemod/public.htm)
– WATSIM (http://www.agp.uni-bonn.de/agpo/rsrch/wats_e.htm)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Elements of a Multi-Commodity Model
• Behavioural functions:defining quantities as function of prices, e.g. demand and supply functions
• Price linkage functions:defining e.g. import prices from border prices and tariffs
• Market balances
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Result as an economic equilibrium
• Marginal willingness to pay = prices paid by consumers(Quantities demanded are on demand function)
• Marginal costs = prices received by producers(Quantities supply are on supply function)
• Markets are cleared “Planned” production equal “Planned demand”
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Solver
World MarketPrices
Flowchart of a Multi-Commodity Model
World MarketBalance
RegionalPrices Pr
SupplySr=f(Pr)
DemandDr=f(Pr)
Net TradeNTr=Sr-Dr
RegionalPrices Pr
SupplySr=f(Pr)
DemandDr=f(Pr)
Net TradeNTr=Sr-Dr
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Components of MultReg
• Final demand– Generalised Leontief Expenditure (GLE) system– Armington assumption with CES functions
• Supply of primary and processed products– Normalised quadratic profit functions– Fat and protein balances for dairies
• Price transmission– Discontinuities (TRQ) solved by fudging functions
• Market balances
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Quantity relations in market model
Production,change in
Intervention StocksExports
Domestic Sales
Demand aggregate(Armington 1)
Cakes,Oils,Dairy
Processing Feed HumanConsumption
Importaggregate
(Armington 2)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Price relations in market model
ProducerPrices
(PPri)
Average priceof quantities consumed
(Arm1P)
Import Prices(Impp)
Average “import” Pricefrom Armington 2
(Arm2P)
Pricefor domesticallyproduced goods
(PMrk)
PSEs,margin
ConsumerPrices
(CPri)
CSEs,margin
Processing marginsfor oilseeds
(ProcMarg)
Processingyields
Processing marginsfor dairy products
(ProcMarg)
Prices for milkfat and protein
(PFatProt)
Importtariffs
(Tars,Tarv)
ExportSubsidies
(Expsub)
TRQs, safeguards
Transportcosts (tcost)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Parameters and Variablesin the Market Module
Scenario parameters Fixed parameters Endogenous Variables• Parameters in behavioural
functions:• Supply• Processing• Human consumption• Feed Use
• Technical parameters:• Crushing yields• Fat & protein content
of milk products
• Prices:• Base year price
producer• Marketing span
for final products• Parameters in functions
determining interventions and subsidized exports
•Demand shifts:• Population growth• GDP development• Changes in
consumption pattern•Shifts in behavioural functions• Exchange rates
Policy instruments:• Administrative prices• Maximal market
interventions• Import Tariffs• Tariff Rate Quotas• Minimal import prices• Subsidised exports
Commitments• Non market PSEs• CSEs
•Quantities:• Supply• Processing• Human consumption• Feed Use• Intervention sales• Bilateral trade flows
•Price elements:• Market prices• Producer price• Consumer price• Processing margins• Import prices• Export subsidies• Tariffs
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Behavioural Functions
• Supply Side:– Supply of primary products– Supply of selected processed products
• Demand Side:– Human consumption– Demand for feed use– Demand of the processing industry
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Processing in the CAPRI Market Model
• Two classes of processed products– Oils and cakes
• Sunflower seed, rape seed, soy beans
• Leontief-Technology assumed
• Supply depends on the value of output (cakes and oils) minus the value of input (oilseed)
– Dairy Submodule• Supply driven by the processing margin of the dairy
• Processing margin:– difference between the retail price and the value of fat and protein
• Fat and protein balances – ensure that all milk components are used up in the dairy
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
13
Functional formsQuantity variable(vriable name)
Functional form(equation name/names)
Driving variables(variable names)
Supply(Production)
Normalized non-symmetric quadratic(ProdNQ_)
Producer prices(PPri)
Supply of cakes and oils (Production)
Leontief(ProcO_)
Processing of oilseeds (Proc),processing yield
Supply of dairy products (Production)
Normalized non-symmetric quadratic(DairyNQ_,ProcMargM_)
Processing margin (ProcMarg) as market price (PPri) minus value of milk fat and protein
Feed(FeedUse)
Normalized non-symmetric quadratic
(FeedNQ_,FeedShift_)
Average price domestic/imports (Arm1P) minus feed subsidiesEnergy shifter (FeedShift, depends on animal production)
Processing(Proc)
Normalized non-symmetric quadratic
(ProcNQ_)
Producer prices (Ppri)exemption: processing margin (ProcMarg) for oilseed processing
Human consumption (Hcon)
Generalised Leontief Expenditure System
Consumer prices (Cpri), income, population
CAPRI CAPRI
Final demand
GLE with Armington
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Final demand: GLE system
Y
YPv
P
YPvX
ii
),(),(
)(
)(),(
PFY
PGYPv
Indirect utility functionF and G functions, homog. of deg. one in prices P,Y = Income
)()()(
)(PFPFY
PG
PGi
i
Use Roy’s identity to derive demands Xi
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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The Generalised Leontief Expenditure function
PopFYG
GDx i
ii
ii
i PDF
i j
jiji PPBG ,
jijjii
i
PPBGP
G,
Expenditure remaining after commitments are covered
iii
DFP
F
Value of minimum commitments
Di = Consumption independent of prices and income
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Final demand: GLE and welfare
)(,
),( simsimsim
refref
simsim
refrefrefsim FY
G
GF
YPv
GFPUe
)(
)(),(
PFY
PGYPv
Indirect utility function
),(
)()(),(
YPv
PGPFPUeY Invert to expenditure function
using U(X) = V(P,Y)
Compute: “How much income would be required at the reference prices to let the consumer reach the Utility Level obtained in the simulation?”
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
18
Why money metric as the utility measurement ?
• Theoretically consistent
• Easy to interprete: income equivalent of the utility in the simulation using the prices of the reference situation
• Can be hence added/compared to costs/revenues/taxes directly to calculate overall welfare (change)
• Becomes part of the objective function(works as „consumer surplus“)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
19
Spatial models
• Bilateral trade streams included• Two standard types:
– Transport cost minimisation– “Armington assumption”:
Quality differences between origins,let consumers differentiate
• We want to allow simultaneous export and import of goods.
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Armington Approach
• Armington, Paul S. 1969"A Theory of Demand for Products Distinguished by Place of Production,“ IMF Staff Papers 16, pp. 159-178.
• CES-Utility aggregatorfor goods consumedfrom different origins
1
,,,,,,
ssrisririri Mx
xi,r Aggregated utility of consuming this productMi,r,s Import streams including domestic sales
shift parameter share parameter parameter related to substitution elasticity
i product,r importing regions, s exporting regions
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
21
First order conditions for the Armington
• First order conditions(FOC) from CES-Utility aggregator( max {U = CES(M1,M2): P1M1+P2M2 = Y} )
• Relation between import streams is depending on:– so called “share parameters”
– multiplied with the inverse import price relation– exponent the substitution elasticity
• Imperfect substitution (“sticky” import shares)
11
1,,
2,,
2,,
1,,
2,,
1,,
rri
rri
rri
rri
rri
rri
P
P
M
M
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
22
Flowchart
RegionalPrices Pr
SupplySr=f(Pr)
DomesticSales
Imports
RegionalPrices Pr
SupplySr=f(Pr)
Domestic Sales
Imports
GLE demandxi,r = f(PCES)
1
11,,1,,,,
rrrirririri Mx
1
11,,1,,,,
rrrirririri Mx
GLE demandxi,r = f(PCES)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
23
Problems of the Armington Approach
• Few empirical estimations of the parameters=> substitution elasticities are set by a “rule-of-thumb”
• A zero stream in the calibrated pointsremains zero in all simulation runs
• The sum of physical streams (domestic sales + imports) is not equal to the utility aggregate in simulations !!!(demand “quantities” are not longer tons, but a utility measurement ...)
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
24
CES function: Iso-utility lines
0
2
4
6
8
10
12
0 2 4 6 8 10 12
Consumption of domestic beef
Con
sum
ptio
n of
impo
rted
bee
f
(M1,M2)
(M1*,M2
*)
),(),( *2
*121 MMxMMx
Enforced in calibration by choice of
21 MM *2
*1 MM
CAPRI CAPRI
Supply of primary and processed products
Normalised quadratic profit function
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Reminder – Micro Theory
Production in implicit form:
Maximizing Profit:
Optimal Supply:
Input Demand:
Normalized Quadratic
Profit Function:
)21( max 2
1110 XXqpXX
012
11
**
5.0)(
),(
ananan
qVpXqp
21110 2
1 XXV
p
qX
Xp
qa
p
qa
1 11
111 11 1
1*
0
2
11
2
1111
21
0*
2
11
2
1
2
1a
q
pa
q
pV
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Processing industry
• Normalised quadratic profit function plus
– Fixed processing yield for oilseed crushing
– Protein and fat balances for dairies
CAPRI CAPRI
Price Transmission
Smoothing out corners with fudging functions
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Motivation
srssr
asrrss
mrks
impsr DTTCSPP ,,,,, 1
Import price is foreign price minus subsidies plus transport costs and tariffs
S = export subsidied of exporting countryC = transportation costTa = ad-valorem tariffTs = specific tariffD = variable import levy to emulate entry price system
Discontinuities:
-If TRQ is filled, MFN tariff is applied, otherwise tariff is lower
-If import price is higher than the min. border price, tariff is lower than MFN
-If import price is higher than the entry price, tariff is also lower than MFN
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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Handling functions with corners
• f = max (0, x) and g = min (x, y) are very difficult for solver because the derivative in the corner is not defined/unique.
• Common approximations: (try x = 10, x = -10) f* = ½(x + (x2 + ) – )g* = ½(x + y – ((x – y)2 + ) – )
• h(x) = {l if x ≤ C, u if x > C} can be approximatedusing logistic function, cumulative normal distribution function or GAMS internal sigmoid() to obtain S-shaped curve.
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
31
Illustration TRQ
• TRQ = Tariff Rate Quota• If import volume is below
quota, tariff < MFN tariff• Bilateral or global• Modelled by GAMS-function
“sigmoid”, represented by f()
T = Tpref + (Tmfn-Tpref)f(M – TRQ) TRQ Import
Tpref
Tmfn
Tariff
True functionSigmoid function
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
32
Illustration minimum border price
• If Pcif is below the minimum border price, a variable levy is added to reach the border price
• The additional levy is limited by the MFN rate
Dtrue = min (max (0,Pcif +Tmfn - Pmin) ,Tmfn)
D = ½(F + Tmfn -((F- Tmfn)2 +2) - )
F = ½(Pcif+Tmfn -Pmin+((Pcif+Tmfn -Pmin)2 +2) - )
Tmfn
Pcif
Pimp
True functionSigmoid function
DPmin
CAPRI CAPRI
Iterative solution
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
34
SupplyRegionaloptimisation
models
Perennialsub-module
Markets Multi-commodityspatial market model
Prices
Reminder – General Model Layout
Quantities
Iterations Comparative Static Equilibrium
Young animal tradeDirect payment model
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
35
On convergence
d
s
q
p
p0 p0
q
ps
d
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
36
Conclusions
• If “demand elasticity” > “supply elasticity”, it will converge, otherwise not
• CAPRI has to be solved iteratively• Elasticities are chosen bases on economic
criteria not to obtain convergence
We will likely need some mechanism promote convergence in CAPRI
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
37
Different ways of promoting convergence
• Adjustment cost: Additional production cost for deviating from the supply in the previous step
• Price expectation: Supply uses weighted average of prices in several previous step. Used in CAPRI
• Partial adjustment: Supply only moves a fraction of the way towards the optimum in each step
• Approximate supply functions used in market instead of fixed supply. Used in CAPRI
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
38
Approximation of supply functions
• The implicit supply function is unknown– Difficult to derive for CAPRI– Has non-differential points (corners) difficult to solve together
with market model
• Assume “any” simple supply function that approximates the supply model
• Calibrate the parameters in each step so that the supply response of last step is reproduced
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
39
Approximating supply
p0
q
ps
d
• Assume the “explosive situation”…
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
40
Approximating supply
p0
q
ps
d
s’s’
q0
• Supply function is unknown (supply is a black box)
• Assume any supply function
• Starting with some price, compute supply
• Calibrate the assumed supply function to that point
• Solve supply + demand simultaneously for new price
• Iterate…
CAPRI CAPRI
Calibration issues
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
42
Calibration of supply parameters
Only one observation of Quantities and (normalized) prices
→ additional information / constraints needed:• Micro Theory:
– Symmetry– Homogeniety– Correct Curvature
• Literature:– Elasticities
ii ikk npbspcsX *
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
43
Parameter calibration
Original elasticities
Restrictions:Micro theory
Constraints of minimisation problem
SymmetryHomogeneityCorrect
Curvature
Objective:keep close
to original ones
Consistent elasticities
Consistent parameters
Functionalform
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
44
Calibration of parametersto given elasticities
• Search parameter vector which produces a regular demand system(here: symmetric pdb with non-negative off-diagonal elements)
• Reproduces the observed combinationof prices and quantities
• And leads to point elasticities „close“ to the given ones
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
45
Point elasticities of the Generalised Leontief Expenditure function
PopFYG
GiPCDDemand i
pp
PopDemand
Y
G
GiPop
Demand
Y
Y
Demand
r
ipYp
,
jiPricePricePbdPbdPrice
GiGij
where
jiPopFYG
GiGi
G
Gij
Price
Demand
ppppppp
ppp
pppp
p
ppp
11,1,21
11,
2
11,
11,
:
Marshallian Demands for any function G and Fand their derivatives versus prices Gi and Fi
Income elasticities of demand
Cross price elasticities of demand
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
46
Regularity conditions I
• Symmetry of second derivatives,here ensured if pdbp,p1 = pdbp1,p1
• Homogeniety of degree one in prices,guaranteed by functions F and G
• Adding up fulfilled, use Eurer‘s law
i
i i
xx
xaxa
)()(
YFFYG
G
PricePCDFYG
PriceGi
PriceDemandp
ppp
pp
ppp
)(
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
47
Regularity conditions II
• And the correct „curvature“, i.e. marginal utility decreasing in quantities is fulfilled if all off-diagonal elements of pdb are non-negative...
• However, then the form does not allow for Hicksian complemetarity (not fully flexible)