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