CAPRI CAPRI CAPRI market model Torbjörn Jansson* Markus Kempen *Corresponding author +49-228-732323 www.agp.uni-bonn.de Department for Economic and Agricultural Policy Bonn University Nussallee 21 53115 Bonn, Germany CAPRI Training Session in Warzaw June 26-30, 2006 CAPRI Common Agricultural Policy Regional Impact
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CAPRI CAPRI market model Torbjörn Jansson* Markus Kempen *Corresponding author +49-228-732323 Department for Economic and Agricultural.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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CAPRI Training Session in Warzaw, June 26-30, 2006
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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
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On convergence
d
s
q
p
p0 p0
q
ps
d
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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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
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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
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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
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Approximating supply
p0
q
ps
d
• Assume the “explosive situation”…
CAPRI CAPRI
CAPRI Training Session in Warzaw, June 26-30, 2006
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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
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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
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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
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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
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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
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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
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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)