The document describes a spatial equilibrium model developed for Costa Rica that includes 17 agricultural products, 6 regions, and an international market region. The model accounts for transport costs between regions, tariffs, and import/export quotas. The purpose is to optimize production, consumption, and transport flows between regions given different supply and demand conditions. A graphical and mathematical model is presented and analyzed using an example involving welfare calculations for two regions with and without trade.

1. Space and Economics
Chapter 10: Spatial Equilibrium Modelling
Author
Rob Schipper (Wageningen, the Netherlands)
April 7, 2010

2. Spatial Equilibrium Modelling
Purpose
Graphical model
Mathematical model
Example SEM in Costa Rica
Advantages & Disadvantages
2

3. Study area: Costa Rica with 6 regions
3

4. Spatial Equilibrium Model for Costa Rica
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
4

5. Purpose of Spatial Equilibrium Model
Different regions within a country:
Production
Consumption
Transport costs between regions
Optimal allocation of:
Production activities
Available produce
Transport flows
5

6. Graphical Model
60
Region 1 Region 2
54
48
42
36
30 p2*
24
p1* 18
12
6
0
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*
6

7. Graphical model: no transport costs
Region 1 Trade Region 2
Price (p2)
Price (p)
60 60 60
Price (p1)
54 54 54
48 48 48
42 42 42
36 36 36
30 30 30
24 24 24
18 18 18
12 12 12
6 6 6
0 0 0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1) Quantity (ED; ES)
Quantity (Q2)
Supply 1 Demand 1 ES 1 + TC ED 2 ES 1 Supply 2 Demand 2
Excess supply and excess demand with welfare consequences:
Consumer welfare Producer welfareTotal welfare
Region 1 loss gain gain
Region 2 gain loss gain
7

8. Welfare function: General format
x1d x1 s x2d x2s
W = W1 + W 2 = ∫ p (x )dx
1 1d 1d − ∫ p (x )dx
1 1s 1s + ∫ p (x )dx
2 2d 2d − ∫ p (x )dx
2 2s 2s
0 0 0 0
Demand and Supply Functions:
p1 = −0.5 x1d + 40,
p1 = x1s + 1,
p2 = −0.25 x2 d + 50,
p2 = 0.5 x2 s + 2.
Welfare function:
W = −0.25 x12d + 40 x1d − 0.5 x12s − x1s − 0.125 x2 d + 50 x2 d − 0.25 x2 s − 2 x2 s
2 2
8

9. Example: Table 10.1
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
Zero transport costs!
9

10. Graphical model: no transport costs
Region 1 Trade Region 2
Price (p2)
Price (p)
60 60 60
Price (p1)
54 54 54
48 48 48
42 42 42
36 36 36
30 30 30
24 24 24
18 18 18
12 12 12
6 6 6
0 0 0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1) Quantity (ED; ES)
Quantity (Q2)
Supply 1 Demand 1 ES 1 + TC ED 2 ES 1 Supply 2 Demand 2
Equilibrium conditions: p1* = p* = p2*
dem1 = sup11 ; dem2 = sup12 + sup22
sup1 = sup11 + sup12 ; sup2 = sup22
p#≥ 0 ; prod# ≥ 0 ; cons# ≥ 0
10

11. Graphical model: with transport costs
Region 1 Trade Region 2
Price (p2)
Price (p)
60 60 60
Price (p1)
54 54 54
48 48 48
42 42 42
36 36 36
30 30 30
24 24 24
18 18 18
12 12 12
6 6 6
0 0 0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quantity (Q1) Quantity (ED; ES) Quantity (Q2)
Supply 1 Demand 1 ES 1 + TC ED 2 ES 1 Supply 2 Demand 2
Consumer welfare Producer welfareTotal welfare
Region 1 loss gain gain
Region 2 gain loss gain
11

12. From Graph to Mathematical model (1)
Regional demand functions:
pdemand = ademand – bdemand * qdemand
Regional supply functions:
Price (p2)
60
psupply = asupply + bsupply * qsupply 48
36
Coefficients a are intercepts 24
Coefficients –b and +b are slopes 12
0
0 3 6 9 12 15 18 21 24
Quantity (x2, y2)
Supply 2 Demand 2
12

13. From Graph to Mathematical model (2)
Quasi welfare function:
Consumer surplus + Producer surplus
=
Price (p2)
60
area below demand curve 48
36
area below supply curve 24
12
0
0 3 6 9 12 15 18 21 24
Quantity (x2, y2)
Supply 2 Demand 2
13

14. From Graph to Mathematical model (3)
The ‘excess supply’ region this configuration differs from the
comparable configuration in ‘excess demand’ region
60 Excess supply 60
Excess demand
54 54
48 48
42 42
36 36
30 30
24 24
18 18
12 12
6 6
0 0
0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0
Quant ity (Q2)
Quant it y (Q1)
Supply 2 Demand 2
Supply 1 Demand 1
14

15. Mathematical model (1)
Maximise total quasi welfare:
 q demand 
q supply
Z = Max ∑  ∫ (a − b q )dq − ∫ (a + b q )dq 
d
i i
d d
i
d
i
s
j
s
j
s
j
s
j
regions i , j  
 0 0 
This is equivalent to:
constant + {aid qid − 1 bid (qid ) 2 }
Z = Max ∑ 
2

regions i , j 
 − {a j q j + 1 b j (q j ) } 
s s
2
s s 2

15

16. Mathematical model (2)
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
16

17. Mathematical model (3)
The Quasi welfare function becomes:
 d d 1 d d 2  s s 1 s s 2
2
( )
max Z = ∑ a j Q j − b j Q j  − ∑  ai Qi − bi (Qi )  − ∑ ∑ t ij Tij
  i  2  j j
j
Subject to constraints: Qd ≤ ∑ Tij
j
(no excess demand)
i
∑T ij ≤ Qis (no excess supply)
j
(non negativity)
Q ≥ 0,Q ≥ 0,Tij ≥ 0
s
i
s
i
P jd = a d − b d Qd
j j j
Pis = ais − bisQis
17

18. Mathematical model (4)
Lagrange function:
 d d 1 d d 2  s s 1 s s 2
2
( )
L = ∑ a j Q j − b j Q j  − ∑  ai Qi − bi (Qi )  − ∑ ∑ t ij Tij
  i  2  j j
j
 
−µ Q j − ∑ Tij 
d
j
d
 i 
 
−µi ∑ Tij − Qi 
s

s

 j 
First order conditions (FOCs)?
18

19. Mathematical model (5)
First order conditions (FOCs):
• With respect to the quantity demanded in region j
∂L
= a d − b d Q d − µ d ≤ 0 all j (1)
∂Q j
d j j j j
• With respect to quantity supplied in region i
∂L
= −ais − bis Qis + µ is ≤ 0 all i (2)
∂Qi s
• With respect to quantity transported from region i to region j
∂L
= −t ij + µ d − µ is ≤ 0 all i and j (3)
∂Tij
j
Using the 1st FOC, in case quantity demanded in region j is non-negative →
µ d = a d − b d Q d = Pjd all j
j j j j
Using the 2nd FOC, in case quantity supplied in region i is non-negative →
µ is = ais + bis Qis = Pi s all i
Then it follows from the 3rd FOC that:
µ d ≤ t ij + µ is , or Pjd ≤ t ij + Pi s
j
Because of the Kuhn-Tucker FOCs, there are two possibilities:
1. Pjd = tij + Pi s → Tij > 0, meaning, that there is (might be) trade between supply region i and demand region j, or
2. Pjd < t ij + Pi s → Tij = 0, meaning, that there is no trade between supply region i and demand region j 19

20. Model with regional supply, demand functions,
and transport between regions
Similar as in Model 8.4 of Hazell & Norton, but with a non-linear (quadratic)
objective function.
Max Z = ∑∑ (α jr ' − 0.5β jr ' D jr ' ) D jr ' − ∑∑ C (Q jr ) − ∑∑∑ ∆ jrr 'T jrr ' (1)
j r j r j r r'
Such that:
∑ T jrr ' ≤ Q jr , all r, j [µ ]
jr (2)
r'
D jr ' ≤ ∑ T jrr ' , all r’, j [µ ' ]
jr ' (3)
r
 a kjr 
∑  y Q jr =∑ a kjr X jr ≤ bkr , all r, k
 
[λkr ] (4)
j  jr  j
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  a kjr 
Qjr = yjr Xjr Supply (= Production) is yield times area Thus:  Q jr = a kjr X jr
y 
From the FOCs, under positive demand (Djr’ > 0) and  jr 
supply (Qjr > 0), two conditions can be derived:
1. µ ' jr ' = α r ' j − β r ' j D jr ' = P r ' j
2. P r ' j ≤ C ′( Q rj ) + ∑ ( a krj / γ rj ) λ rk + ∆ jrr '
k 20
What do they mean?

21. Example of Spatial Equilibrium Modelling
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
21

22. Methodology (1)
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
22

23. Study Area: 6 Regions of Costa Rica
23

24. Methodology (2)
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
24

25. Spatial Equilibrium Model: Wrap Up
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)
25

26. Advantages & Disadvantages
Optimal allocation of production
Optimal transport flows
Evaluate effect of, for example:
Infrastructure development
Technological progress
Trade liberalisation
Demographic changes
26

27. 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