Chapter 3: Classical Location Theory of the Firm

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Chapter 3: Classical Location Theory of the Firm

  1. 1. Space and Economics Chapter 3: Classical Location Theory of the Firm Author Wim Heijman (Wageningen, the Netherlands) July 21, 2009
  2. 2. 3. Classical location theory of the firm 3.1 Minimization of transportation costs: one final product 3.2 Minimization of transportation costs: one resource, one final product 3.3 Trans shipment costs 3.4 Other location factors 3.5 Alfred Weber’s theory on location of the firm 3.6 The Theory of the market areas 3.7 Spatial elasticity of demand 3.8 Market forms: spatial duopoly 3.9 Application
  3. 3. 3.1 Minimization of transportation costs: one final product Location of one ice cream vendor on the beach Customers equally distributed over the beach Customers have equal preferences for ice cream The lower the average distance between ice cream vendor and customers the more ice cream will be sold
  4. 4. 3.1 Minimization of transportation costs: one final product 100m 100m 100m 100m A B C D E Figure 3.1: Beach with five visitors.
  5. 5. 3.1 Minimization of transportation costs: one final product 0 m + 100 m + 200 m + 300 m + 400 m A: = 200 m, 5 100 m + 0 m + 100 m + 200 m + 300 m B: = 140 m, 5 200 m + 100 m + 0 m + 100 m + 200 m C: = 120 m, 5 300 m + 200 m + 100 m + 0 m + 100 m D: = 140 m, 5 400 m + 300 m + 200 m + 100 m + 0 m E: = 200 m. 5
  6. 6. 3.1 Minimization of transportation costs: one final product Average 210 distance 200 190 180 170 160 150 140 130 120 110 A B C E D Location
  7. 7. 3.1 Minimization of transportation costs: one final product 3 Figure 3.3: Optimum location in a two dimensional space. 2 b a 1 A B C
  8. 8. 3.2 Minimization of transportation costs: one resource, one final product ta tb A B S 100 km Figure 3.4: Location of a firm that produces only one product with the help of one raw material
  9. 9. 3.2 Minimization of transportation costs: one resource, one final product K = ata f a + btb f b t b = 100 − t a , a :1000; b :100; f a = fb = 0.10 K = 0.10 ⋅1000t a + 100 ⋅ (100 − t a ) ⋅ 0.10 = 1000 + 90t a .
  10. 10. 3.2 Minimization of transportation costs: one resource, one final product K 11 10 9 8 7 K = 1000 + 90 x ta 6 x €1000 5 4 3 2 1 0 10 20 30 40 50 60 70 80 90 100 km A B ta Figure 3.5: Minimization of transportation costs with one input and one product.
  11. 11. 3.3 Trans shipment costs * Trans shipment costs are costs that are made when the transportation mode changes. * For example, in a sea port, the cargo may be transported further to the hinterland by truck, rail or inland waterways. * Trans shipment costs are normally expressed in money units per weight unit (e.g. euro’s per ton)
  12. 12. 3.3 Trans shipment costs S O G M tg tm T Figure 3.6: Location S of a business with transshipment location O
  13. 13. 3.3 Trans shipment costs K = gt g f g + mt m f m + mom + gog , tm = T − t g , K = ( gf g − mf m )t g + mf mT + mog + gom .
  14. 14. 3.3 Trans shipment costs In the case of trans shipment costs the optimum location is found with: K g = mf mT + mom , K m = gf gT + gog , K o = ( gf g − mf m )t g + mf mT .
  15. 15. 3.3 Trans shipment costs If K g = Km = Ko. Then the firm is footloose
  16. 16. 3.4 Other location factors Apart from transportation costs there are two other important location factors: labour costs; agglomeration benefits or external economies of scale.
  17. 17. 3.4 Other location factors Agglomeration: a spatial clustering of interacting firms that is mutually beneficial because it generates a decrease in production costs Deglomeration: spatial deconcentration of firms because of external diseconomies of scale
  18. 18. 3.5 Alfred Weber’s theory on location y yc C tc ys S tb ta xb A xc xs B x Figure 3.8: Location triangle.
  19. 19. Alfred Weber (1886 1958)
  20. 20. Pierre Varignon (1654 1722)
  21. 21. 3.5 Alfred Weber’s theory on location C tc S tb ta A B Figure 3.9: The Varignon frame.
  22. 22. 3.5 Alfred Weber’s theory on location Original Varignon Frame
  23. 23. 3.5 Alfred Weber’s theory on location t a = xs2 + y s2 , tb = y s2 + ( xb − xs ) 2 , t c = ( xs − xc ) 2 + ( yc − y s ) 2 . K = at a f a + bt b f b + ct c f c , K = af a xs2 + y s2 + bf b y s2 + ( xb − xs ) 2 + cf c ( xs − xc ) 2 + ( yc − y s ) 2 . ∂K ∂K = = 0. ∂x s ∂y s
  24. 24. 3.5 Alfred Weber’s theory on location +60 -70 V +40 +20 S -10 W Figure 3.10: Transportation cost optimum with isodapanes.
  25. 25. 3.5 Alfred Weber’s theory on location +60 +60 +40 +40 +20 +20 S1 S S2 -60 Figure 3.11: Agglomeration benefits.
  26. 26. 3.5 Alfred Weber’s theory on location y 30 20 +60 +40 10 +20 Spatial margin to profitability 30 20 10 0 10 20 30 x y =0 10 20 30 100 100 Total Revenue Spatial 80 80 Cost curve 60 60 40 40 20 20 Production costs y =0 30 20 10 0 10 20 30 Figure 3.12: Spatial margins to profitability. Figure 3.12: Spatial margins to profitability.
  27. 27. 3.5 Alfred Weber’s theory on location y yc 40 C y =40 y =35 y =30 tc y =25 y =20 y =15 ys S y =10 ta tb y=5 xb y =0 A xc xs B x 20 60 Figure 3.13: Sections in the location triangle.
  28. 28. 3.5 Alfred Weber’s theory on location y =40 K 110 y =35 105 y =30 100 y =25 y =20 95 y =15 90 y =10 y =5 y =0 85 80 76,44 75 70 0 10 20 30 40 50 60 x 5 15 25 35 45 55 Figure 3.14: Transportation cost curves of a classical location problem.
  29. 29. 3.5 Alfred Weber’s theory on location K = 0.5 xs2 + y s2 + y s2 + (60 − xs ) 2 + ( xs − 20) 2 + (40 − y s ) 2 . K Figure 3.15: 3 D presentation of the transportation cost function.
  30. 30. www.corusgroup.com/file_source/staticfiles/corus_locations.pdf 3.5 Alfred Weber’s theory on location
  31. 31. 3.6 The theory of market areas Three important authors: Walter Christaller (1934), Tord Palander (1935), August Lösch (1939).
  32. 32. 3.6 The theory of market areas
  33. 33. 3.6 The theory of market areas Market area A Market area B Prices and Pa costs Pb Fa Fb O A G B Z Distance Figure 3.17: Palander’s market areas
  34. 34. 3.6 The theory of market areas quantity distance B A distance O distance distance Figure 3.18: Quantity consumed as a function of the distance to the location of the producer (O); the spatial demand function.
  35. 35. 3.6 The theory of market areas Figure 3.19: Seven firms with their market areas.
  36. 36. 3.6 The theory of market areas Figure 3.20: Hexagonal structure and hierarchy of central places.
  37. 37. 3.7 Spatial elasticity of demand ∆q % change in demand q ∆q x dq x Es = d = = ≈ . % change in distance ∆x ∆x q dx q x −α q = Kx . −α −1 − αKx x E = d s −α = −α . Kx
  38. 38. 3.7 Spatial elasticity of demand 12 10 8 q 6 4 2 0 1 3 5 7 9 11 13 15 17 19 x Figure 3.21: Spatial demand curve with fixed spatial demand elasticity of 1.
  39. 39. 3.8 Market forms: spatial duopoly 1 A B 2 B A 3 A B 4 B A 5 A B Figure 3.22: Spatial duopoly: Hotelling’s Law.
  40. 40. 3.8 Market forms: spatial duopoly Hotelling’s Law: spatial competition leads to clustering of competitors in the centre Hotelling’s Law is based on a zero spatial demand elasticity: In terms of game theory Hotelling’s Law describes a Nash Equilibrium α = 0. Hotelling’s Law is also used by political scientists to explain the positioning of candidates running for a political position: http://www.rawstory.com/exclusives/steinberg/ice_cream_08 2005.htm
  41. 41. Harold Hotelling (1895 1973)
  42. 42. 3.9 Application www.sugartech.co.za/factories/index.php Sugar factory Figure 3.23: Location of sugar factories in the Netherlands in 1992

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