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Space and Economics
Chapter 10: Spatial Equilibrium Modelling
Author
Rob Schipper (Wageningen, the Netherlands)
April 7, 2010
Spatial Equilibrium Modelling
� Purpose
� Graphical model
� Mathematical model
� Example SEM in Costa Rica
� Advantages & Disadvantages
2
Study area: Costa Rica with 6 regions
3
� Spatial Equilibrium Model includes:
� 17 of the major agricultural products
� 6 planning regions of Costa Rica
� International market as 7th region
� Transport costs between the 7 regions
� Tariffs on import and export prices
� Import and export quota
Spatial Equilibrium Model for Costa Rica
4
� Different regions within a country:
� Production
� Consumption
� Transport costs between regions
� Optimal allocation of:
� Production activities
� Available produce
� Transport flows
Purpose of Spatial Equilibrium Model
5
Graphical Model
0
6
12
18
24
30
36
42
48
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quant it y (Q1)
Supply 1 Demand 1
Supply from region 1 to region 2 when p > p1*
Demand from region 2 from region 1 at p < p2*
p1*
p2*
Region 1 Region 2
6
06
121824303642485460
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q2)
Pri
ce (
p2
)
Supply 2 Demand 2
0
612
1824
303642
48
5460
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (ED; ES)P
rice
(p)
ES 1 + TC ED 2 ES 1
0
612
18
24
30
36
4248
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1)
Pric
e (p
1)
Supply 1 Demand 1
Region 1 Region 2
Excess supply and excess demand with welfare consequences:
Consumer welfare Producer welfareTotal welfareRegion 1 loss gain gainRegion 2 gain loss gain
Trade
Graphical model: no transport costs
7
W = W1 +W2 = p10
x1d
∫ x1d( )dx1d − p10
x1s
∫ x1s( )dx1s + p2 x2d( )0
x2d
∫ dx2d − p2 x2s( )0
x2s
∫ dx2s
.25.0
,5025.0
,1
,405.0
22
22
11
11
+=+−=
+=+−=
s
d
s
d
xp
xp
xp
xp
Demand and Supply Functions:
ssddssdd xxxxxxxxW 2222
221
211
21 225.050125.05.04025.0 −−+−−−+−=
Welfare function:
Welfare function: General format
8
Example: Table 10.1
Zero transport costs!
Regime Concept Region 1 Region 2 Total
No trade Welfare (=CS+PS) 507.00 1536.00 2043.00
Consumer surplus 169.00 512.00 681.00
Producer surplus 338.00 1024.00 1362.00
Trade Welfare (=CS+PS) 539.67 1552.33 2092.00
Consumer surplus 69.44 672.22 741.67
Producer surplus 470.22 880.11 1350.33
Differences Δ Welfare (=CS+PS) 32.67 16.33 49.00
Δ Consumer surplus @99.56 160.22 60.67
Δ Producer surplus 132.22 @143.89 @11.67
9
Graphical model: no transport costs
06
121824303642485460
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q2)
Pri
ce (
p2
)
Supply 2 Demand 2
0
612
1824
303642
48
5460
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (ED; ES)P
rice
(p)
ES 1 + TC ED 2 ES 1
0
612
18
24
30
36
4248
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1)
Pric
e (p
1)
Supply 1 Demand 1
Region 1 Region 2Trade
Equilibrium conditions: p1* = p* = p2
*
dem1 = sup11 ; dem2 = sup12 + sup22sup1 = sup11 + sup12 ; sup2 = sup22p#≥ 0 ; prod# ≥ 0 ; cons# ≥ 0
10
0
6
12
18
24
3036
42
48
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q2)
Pric
e (p
2)
Supply 2 Demand 2
0
612
1824
30
3642
48
5460
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (ED; ES)
Pric
e (p
)
ES 1 + TC ED 2 ES 1
0
6
12
18
24
3036
42
48
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1)
Pric
e (p
1)
Supply 1 Demand 1
Region 1 Region 2
Consumer welfare Producer welfareTotal welfareRegion 1 loss gain gainRegion 2 gain loss gain
Trade
Graphical model: with transport costs
11
0
12
24
36
48
60
0 3 6 9 12 15 18 21 24
Quantity (x2, y2)
Pric
e (p
2)Supply 2 Demand 2
� Regional demand functions:
pdemand = ademand – bdemand * qdemand
� Regional supply functions:
psupply = asupply + bsupply * qsupply
� Coefficients a are intercepts
� Coefficients –b and +b are slopes
From Graph to Mathematical model (1)
12
0
12
24
36
48
60
0 3 6 9 12 15 18 21 24
Quantity (x2, y2)P
rice
(p2)
Supply 2 Demand 2
Quasi�welfare function:
Consumer surplus + Producer surplus
=
area below demand curve
@
area below supply curve
From Graph to Mathematical model (2)
13
From Graph to Mathematical model (3)
0
6
12
18
24
30
36
42
48
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quant it y (Q1)
Supply 1 Demand 1
0
6
12
18
24
30
36
42
48
54
60
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0Quant ity (Q2)
Supply 2 Demand 2
The ‘excess supply’ region this configuration differs from the comparable configuration in ‘excess demand’ region
Excess supply Excess demand
14
� Maximise total quasi@welfare:
� This is equivalent to:
∑ ∫∫
+−−=
ji
qsj
sj
sj
sj
qdi
di
di
di dqqbadqqbaMaxZ
, regions 00
supplydemand
)()(
∑
+−
−+=
jisj
sj
sj
sj
di
di
di
di
qbqa
qbqaMaxZ
, regions2
21
221
})({
})({constant
Mathematical model (1)
15
� Transport costs between supply region i and demand region j:
� unit transport costs tij� transport flow Tij
� total transport costs tij * Tij
� Transport costs are a cost to society
Mathematical model (2)
16
Mathematical model (3)
maxZ = a jdQj
d −12
b jd Qj
d( )2
j
∑ − aisQi
s −12
bis Qi
s( )2
i
∑ − tij Tijj
∑j
∑
The Quasi@welfare function becomes:
Subject to constraints: Qjd ≤ Tij
i
∑
Tijj
∑ ≤ Qis
Qis ≥ 0,Qi
s ≥ 0,Tij ≥ 0
Pjd = a j
d − bjdQj
d
Pis = ai
s − bisQi
s
(no excess demand)
(no excess supply)
(non negativity)
17
L = a jdQj
d −12
b jd Qj
d( )2
j
∑ − aisQi
s −12
bis Qi
s( )2
i
∑ − tij Tijj
∑j
∑
−µ jd Qj
d − Tiji
∑
−µis Tij
j
∑ − Qis
Mathematical model (4)
Lagrange function:
First order conditions (FOCs)?
18
• With respect to the quantity demanded in region j
0≤−−=∂∂ d
jdj
dj
djd
j
QbaQ
L µ all j (1)
• With respect to quantity supplied in region i
0≤+−−=∂∂ s
isi
si
sis
i
QbaQ
L µ all i (2)
• With respect to quantity transported from region i to region j
0≤−+−=∂∂ s
idjij
ij
tT
L µµ all i and j (3)
Using the 1st FOC, in case quantity demanded in region j is non-negative → dj
dj
dj
dj
dj PQba =−=µ all j
Using the 2nd FOC, in case quantity supplied in region i is non-negative → s
isi
si
si
si PQba =+=µ all i
Then it follows from the 3rd FOC that: siij
dj t µµ +≤ , or s
iijdj PtP +≤
Because of the Kuhn-Tucker FOCs, there are two possibilities: 1. s
iijdj PtP += → Tij > 0, meaning, that there is (might be) trade between supply region i and demand region j, or
2. siij
dj PtP +< → Tij = 0, meaning, that there is no trade between supply region i and demand region j
First order conditions (FOCs):
Mathematical model (5)
19
Model with regional supply, demand functions,
and transport between regionsSimilar as in Model 8.4 of Hazell & Norton, but with a non-linear (quadratic) objective function. Max ∑∑∑∑∑∑∑ ∆−−−=
j r rjrrjrr
j rjr
jjrjrjrjr
r
TQCDDZ'
'''''' )()5.0( βα (1)
Such that:
∑ ≤'
'r
jrjrr QT , all r, j [ ]jrµ (2)
∑≤r
jrrjr TD '' , all r’ , j [ ]'' jrµ (3)
∑∑ ≤=
jkrjrkjr
jjr
jr
kjr bXaQy
a , all r, k [ ]krλ (4)
All Qjr, Djr and Tjrr’ ≥ 0 (5)
Djr’ Demand for commodity j in region r’ Qjr Supply of commodity j in region r (with supply = production) Tjrr’ Transport of commodity j from region r to region r’ Xjr Production area with commodity i in region r Qjr = yjr Xjr Supply (= Production) is yield times area
jrkjrjr
jr
kjr XaQy
a=
From the FOCs, under positive demand (Djr’ > 0) and supply (Qjr > 0), two conditions can be derived: 1. P= D jrjrjrjrjr '''''' βαµ −=
2. '' jrrrkrjkrjk
rjjr )/a()Q(C P ∆++′≤ ∑ λγ
What do they mean?
Thus:
20
� Development of a methodology to:
� Model spatial patterns of supply, demand, trade flows and prices of major agricultural products in Costa Rica
� Assessing the degree to which current trade policies (e.g.,import duties and export tariffs) lead to sub@optimal welfare levels
Example of Spatial Equilibrium Modelling
21
� Spatial Equilibrium Model includes:
� 17 of the major agricultural products
� 6 planning regions of Costa Rica
� International market as 7th region
� Transport costs between the 7 regions
� Tariffs on import and export prices
� Import and export quota
Methodology (1)
22
Study Area: 6 Regions of Costa Rica
23
� Model requirements:
� Estimations of supply and demand elasticities
� Production and consumption levels in base year
� Transport costs estimations
� Domestic prices in base year
� World market prices
� Import and export quota levels
Methodology (2)
24
� Objective function:
+ producer surplus
+ consumer surplus
@ transport costs between regions
(for concerned products and regions)
� Restrictions:
� Supply
� Demand
� Export and import limitations, if any (open economy)
� Resources (sometimes added in practice)
Spatial Equilibrium Model: Wrap Up
25
Advantages & Disadvantages
� Optimal allocation of production
� Optimal transport flows
� Evaluate effect of, for example:
� Infrastructure development
� Technological progress
� Trade liberalisation
� Demographic changes
26
Advantages & Disadvantages
� Model difficult to solve for non@linear or non@quadratic welfare function
� No cross price elasticities
� No adjustment costs
� Exogenous transport costs
27