Space and Economics
Chapter 10: Spatial Equilibrium Modelling

Author
Rob Schipper (Wageningen, the Netherlands)

April 7,...
Spatial Equilibrium Modelling


  Purpose
  Graphicalmodel
  Mathematical model
  Example SEM in Costa Rica
  Advanta...
Study area: Costa Rica with 6 regions




3
Spatial Equilibrium Model for Costa Rica

      Spatial   Equilibrium Model includes:
         17  of the major agricult...
Purpose of Spatial Equilibrium Model

  Different   regions within a country:
      Production
      Consumption

  Tr...
Graphical Model

                 Region 1                             Region 2


                                        ...
Graphical model: no transport costs
      Region 1                 Trade                    Region 2




Excess supply and...
Welfare function: General format


Demand and Supply Functions:




Welfare function:



 8
Example: Table 10.1
Regime     Concept               Region 1      Region 2     Total
No trade   Welfare (=CS+PS)         ...
Graphical model: no transport costs
      Region 1                     Trade                       Region 2




Equilibriu...
Graphical model: with transport costs

     Region 1                  Trade              Region 2




         Consumer we...
From Graph to Mathematical model (1)
  Regional      demand functions:
     pdemand = ademand – bdemand * qdemand

  Reg...
From Graph to Mathematical model (2)
Quasi-welfare function:

Consumer surplus + Producer surplus
=
area below demand curv...
From Graph to Mathematical model (3)
The ‘excess supply’ region this configuration differs from the
comparable configurati...
Mathematical model (1)

    Maximise total quasi-welfare:




    This is equivalent to:




15
Mathematical model (2)
    Transport costs between supply region i and demand
     region j:
        unit transport cost...
Mathematical model (3)
The Quasi-welfare function becomes:




Subject to constraints:               (no excess demand)

 ...
Mathematical model (4)
Lagrange function:




First order conditions (FOCs)?

18
Mathematical model (5)
First order conditions (FOCs):




   19
Model with regional supply, demand functions,
     and transport between regions

                                        ...
Example of Spatial Equilibrium Modelling
    Development of a methodology to:

         Model spatial patterns of supply...
Methodology (1)
    Spatial Equilibrium Model includes:
        17 of the major agricultural products
        6 plannin...
Study Area: 6 Regions of Costa Rica




23
Methodology (2)

    Model requirements:
        Estimations of supply and demand elasticities
        Production and c...
Spatial Equilibrium Model: Wrap Up
    Objective function:
         + producer surplus
         + consumer surplus
      ...
Advantages & Disadvantages

    Optimal allocation of production
    Optimal transport flows
    Evaluate effect of, fo...
Advantages & Disadvantages

    Model difficult to solve for non-linear or non-quadratic
     welfare function
    No cr...
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Chapter 10: Spatial Equilibrium Modelling

  1. 1. Space and Economics Chapter 10: Spatial Equilibrium Modelling Author Rob Schipper (Wageningen, the Netherlands) April 7, 2010
  2. 2. Spatial Equilibrium Modelling   Purpose   Graphicalmodel   Mathematical model   Example SEM in Costa Rica   Advantages & Disadvantages 2
  3. 3. Study area: Costa Rica with 6 regions 3
  4. 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. 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. 6. Graphical Model Region 1 Region 2 p2* p1* Supply from region 1 to region 2 when p > p1* Demand from region 2 from region 1 at p < p2* 6
  7. 7. Graphical model: no transport costs Region 1 Trade Region 2 Excess supply and excess demand with welfare consequences: Consumer welfare Producer welfare Total welfare Region 1 loss gain gain Region 2 gain loss gain 7
  8. 8. Welfare function: General format Demand and Supply Functions: Welfare function: 8
  9. 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. 10. Graphical model: no transport costs Region 1 Trade Region 2 Equilibrium conditions: p1* = p* = p2* dem1 = sup11 ; dem2 = sup12 + sup22 sup1 = sup11 + sup12 ; sup2 = sup22 p#≥ 0 ; prod# ≥ 0 ; cons# ≥ 0 10
  11. 11. Graphical model: with transport costs Region 1 Trade Region 2 Consumer welfare Producer welfare Total welfare Region 1 loss gain gain Region 2 gain loss gain 11
  12. 12. From Graph to Mathematical model (1)   Regional demand functions: pdemand = ademand – bdemand * qdemand   Regional supply functions: psupply = asupply + bsupply * qsupply   Coefficients a are intercepts   Coefficients –b and +b are slopes 12
  13. 13. From Graph to Mathematical model (2) Quasi-welfare function: Consumer surplus + Producer surplus = area below demand curve - area below supply curve 13
  14. 14. From Graph to Mathematical model (3) The ‘excess supply’ region this configuration differs from the comparable configuration in ‘excess demand’ region Excess supply Excess demand 14
  15. 15. Mathematical model (1)   Maximise total quasi-welfare:   This is equivalent to: 15
  16. 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. 17. Mathematical model (3) The Quasi-welfare function becomes: Subject to constraints: (no excess demand) (no excess supply) (non negativity) 17
  18. 18. Mathematical model (4) Lagrange function: First order conditions (FOCs)? 18
  19. 19. Mathematical model (5) First order conditions (FOCs): 19
  20. 20. Model with regional supply, demand functions, and transport between regions The SEM model on Sheets 18 & 19 is comparable to Model 8.4 of Lecture 11 Thus: 20
  21. 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. 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. 23. Study Area: 6 Regions of Costa Rica 23
  24. 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. 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. 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. 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
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