Chapter 4: Modern Location Theory of the Firm

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Chapter 4: Modern Location Theory of the Firm

  1. 1. Space and Economics Chapter 4: Modern Location Theory of the Firm Author Wim Heijman (Wageningen, the Netherlands) July 23, 2009
  2. 2. 4. Modern location theory of the firm 4.1 Neoclassical location theory 4.2 The neoclassical optimization problem in a two dimensional space 4.3 Growth poles 4.4 Core and periphery 4.5 Agglomeration and externalities 4.6 Market forms: spatial monopoly 4.7 Spatial duopoly: Hotelling’s Law generalised 4.8 Optimum location from a welfare viewpoint
  3. 3. 4.1 Neoclassical location theory In the Weber model substitution of input factors is not possible: Leontief production function In neoclassical analysis of the locational problem of the firm, substitutability of production inputs is assumed: e.g. Cobb Douglas production function.
  4. 4. 4.1 Neoclassical location theory T t l tg 0 100 L G V Figure 4.1: Location of a firm along a line
  5. 5. 4.1 Neoclassical location theory MAX q = l α g 1− α , ( ) ( ) ( ) ( ) s.t. B = pl + ptl tl l + pg + ptg t g g = pl + ptl tl l + pg + ptg (T − tl ) g.
  6. 6. 4.1 Neoclassical location theory αB l= , 0 ≤ tl ≤ 100. pl + p tl t l g= (1 − α )B , so : pg + pt (T − tl ) g α 1−α  αB q=     (1 − α )B   .  p + plt   p + p g (T − t )   l t l   g t l 
  7. 7. 4.1 Neoclassical location theory Assume:α = 0.5, T = 100, B = 500, ptl = 0.1, p g = 5, pl = 2, pt = 0.2. g Then: 250 250 l= ,g= , so : 2 + 0.1tl 5 + 0.2(100 − tl ) 0 .5 0 .5  250   250  62,500 q=      5 + 0.2(100 − t )   = .  2 + 0.1tl − 0.02tl + 2.1tl + 50 2   l 
  8. 8. 4.1 Neoclassical location theory Table 4.1: Inputs and production along a line. t l g q l 0 125.00 10.00 35.36 10 83.33 10.87 30.10 20 62.50 11.91 27.28 30 50.00 13.16 25.65 40 41.67 14.71 24.75 50 35.71 16.67 24.40 60 31.25 19.23 24.52 70 27.78 22.73 25.13 80 25.00 27.78 26.35 90 22.73 35.71 28.49 100 20.83 50.00 32.27
  9. 9. 4.1 Neoclassical location theory 40 35 30 25 Production q 20 15 10 5 0 L 0 10 20 30 40 50 60 70 80 90 100 G Distance from L: tl Figure 4.2: Spatial production curve.
  10. 10. 4.2 The neoclassical optimization problem in a two dimensional space Min K = ( pa + f ata )a + ( pb + f btb )b + f ctc c, with respect to a, b, xs and ys , s.t. c(a, b) = c* , and : ta = ys2 + xs2 , tb = ys2 + ( xb − xs ) 2 , tc = ( xs − xc ) 2 + ( yc − ys ) 2 ,
  11. 11. 4.2 The neoclassical optimization problem in a two dimensional space This can be solved in two steps: 1. Determine the optimum a and b for given ta, tb, and tc; 2. determine the optimum xs and ys given the solution for a and b. Step 1: ( ) min L = ( p a + f a t a ) a + ( p b + f b t b )b + t c f c c( a, b) − λ c( a, b) − c * , ∂c / ∂a pa + f ata = . K = K (ta , tb , tc ). ∂c / ∂b pb + f btb
  12. 12. 4.2 The neoclassical optimization problem in a two dimensional space Step 2: Because: ta = ys2 + xs2 , tb = ys2 + ( xb − xs ) 2 , tc = ( xs − xc ) 2 + ( yc − ys ) 2 , we can now find the optimum with: ∂K ∂K ∂ta ∂K ∂tb ∂K ∂tc = + + = 0, ∂xs ∂ta ∂xs ∂tb ∂xs ∂tc ∂xs and: ∂K ∂K ∂ta ∂K ∂tb ∂K ∂tc = + + = 0. ∂ys ∂ta ∂ys ∂tb ∂ys ∂tc ∂ys
  13. 13. 4.2 The neoclassical optimization problem in a two dimensional space K 320 310 300 y=40 290 y=35 280 y=30 270 y=25 260 y=20 250 y=15 240 y=10 230 y=5 220 y=0 210 200 190 0 5 10 15 20 25 30 35 40 45 50 55 60 x Figure 4.3: Spatial costs curves in the neoclassical model.
  14. 14. 4.2 The neoclassical optimization problem in a two dimensional space K Figure 4.4: 3 D presentation of the neoclassical cost function.
  15. 15. 4.3 Growth poles A growth pole is a geographical concentration of economic activities Growth Pole is more or less identical with: ‘agglomeration’ and ‘cluster’ 4 types of growth poles: technical, income, psychological, planned growth pole
  16. 16. 4.3 Growth poles Technical growth pole: geographically concentrated supply chain based on forward and backward linkages. Product Chain Semi Finished Product Semi Finished Product Firm A Firm B Firm C Backward Linkage Forward Linkage
  17. 17. 4.3 Growth poles Income growth pole: location of economic activities generates income which positively influences the local demand for goods and services through a multiplier process, also called trickling down effect.
  18. 18. 4.3 Growth poles Psychological growth pole: the image of a region is important. Location of an important industry in a backward region may generate a positive regional image stimulating others to locate in the area.
  19. 19. 4.3 Growth poles Planned growth pole: Government may try to stimulate regional economic development for example by a policy of locating governmental agencies in backward regions.
  20. 20. 4.3 Growth poles Technical Growth Pole Psychological Income Growth Pole Growth Pole Planned Growth Pole Figure 4.6: Types of growth poles.
  21. 21. 4.4 Core and periphery Gunnar Myrdal (1898 1987): Core periphery theory: economic growth inevitably leads to regional economic disparities.
  22. 22. 4.4 Core and periphery Economic growth is geographically concentrated in certain regions (the core) In the core regions polarisation plays an important role. Myrdal calls that “cumulative causation”
  23. 23. 4.4 Core and periphery The core regions attract production factors (labour, capital) from the periphery: “backwash effects” If the cumulative causation continues, congestion appears in the core regions (traffic jams, high land prices, high rents, high wages, etcetera). This will generate migration of land intensive and labour intensive industries from the core to areas outside: “spread effect”. In most cases, areas close to the core profit most from this effect: “spill over areas”.
  24. 24. 4.4 Core and periphery Alfred Weber’s theory on location Technical Location of polarisation a pull element Expansion of Growth of production of employment and Psychological goods and services income: polarisation for the local market income polarisation Increase of local Improvement of tax revenues infrastructure Figure 4.7: The principle of cumulative causation
  25. 25. Gunnar Myrdal (1898 1987)
  26. 26. 4.5 Agglomeration and externalities Economies of scale: costs per unit product decrease if the scale of production increases Two types of externalities: internal; external. Internal economies of scale take place within a firm external economies of scale, a form of externalities, take place between firms External economies of scale may arise in a cluster or agglomeration Figure 3.12: Spatial margins to profitability.
  27. 27. 4.5 Agglomeration and externalities K s = K s ( N s ), K s , N s ≥ 0, dK s dK s dK s < 0, if N s < N s* , > 0, if N s > N s* , = 0, if N s = N s* , dN s dN s dN s K s = αN − β N s + γ , 2 s α , β , γ > 0.
  28. 28. 4.5 Agglomeration and externalities K2 K1 O A B C 2 1 N1 N2 N Figure 4.8: Stable spatial equilibrium.
  29. 29. 4.5 Agglomeration and externalities E K2 K1 D O A B C 2 1 N1 N2 N Figure 4.9: Unstable spatial equilibrium.
  30. 30. 4.5 Agglomeration and externalities K s = αN s2 − βN s + γ , dK s β = 2αN s − β = 0, so : N s = * . dN s 2α N 2αN m = *= * . Ns β
  31. 31. 4.5 Agglomeration and externalities
  32. 32. 4.5 Agglomeration and externalities http://www.liof.com/?id=28 www.emcc.eurofound.eu.int/automotivemap
  33. 33. 4.6 Market forms: spatial monopoly q ( x ) = K | x − x s |− α , 0 ≤ x ≤ xT 0 <α <1 q(x) MSP 0 xT xs x Figure 4.12: Spatial demand curve.
  34. 34. 4.6 Market forms: spatial monopoly xs xT −α −α Q( x) = ∫ K ( x s − x) dx + ∫ K ( x − x s ) dx. 0 xs xT xs = . 2
  35. 35. 4.7 Spatial duopoly: Hotelling’s Law generalised q1(x) q1(x) q1(x) q2(x) q2(x) q2(x) MSP1 MSP2 0 x1 0.5(x 1 + x 2 ) x2 xT x Figure 4.13: Spatial duopoly with two mobile selling points (MSP).
  36. 36. 4.7 Spatial duopoly: Hotelling’s Law generalised 1 ( x1 + x 2 ) x1 2 −α −α Q1 ( x) = ∫ K ( x1 − x) dx + ∫ K (x − x )1 dx, 0 x1 x2 xT −α −α Q2 ( x) = 1 ∫ K ( x2 − x) dx + ∫ K ( x − x2 ) dx. x2 ( x1 + x 2 ) 2
  37. 37. 4.7 Spatial duopoly: Hotelling’s Law generalised The cooperative solution : 1 3 x1 = xT , x2 = xT . 4 4
  38. 38. 4.7 Spatial duopoly: Hotelling’s Law generalised competitive solution:  α 1   2 1 x1 =   xT , 1  1 + 2α2    1   2 + 2α 1 x2 = xT − x1 =  1  xT  1 + 2α 2   The competitive solution represents a so called Nash equilibrium.
  39. 39. 4.7 Spatial duopoly: Hotelling’s Law generalised 1 3 If α → ∞, then x1 → xT and x2 → xT , which is equal to the cooperative 4 4 (efficient) solution. 1 If α → 0, then x1 , x2 → xT , which is the Hotelling Law (Section 3.7). 2 1 1 1 3 For 0 < α < ∞, xT < x1 < xT , and xT < x2 < xT . 4 2 2 4
  40. 40. 4.8 Optimum location from a welfare viewpoint In case of monopolistic competition the products offered are almost perfect substitutes for another For example, restaurants may offer exactly the same meals, but on different locations. Everything else being equal, one prefers a meal in a restaurant on a location which is close by to a meal in a restaurant far away.
  41. 41. 4.8 Optimum location from a welfare viewpoint Figure 4.14: Six restaurants in a circular space.
  42. 42. 4.8 Optimum location from a welfare viewpoint 1 1 d= D 2N The cost per unit distance equals t, so the total transportation costs Ctransport for L customers equal: tL Ctransport = D. 2N
  43. 43. 4.8 Optimum location from a welfare viewpoint With constant marginal costs M and fixed costs per restaurant F, and Q meals, the costs Cmeals of the meals are: Cmeals = NF + MQ. If there is one meal per customer per day, then, with L customers and N restaurants, total costs per day Cmeals for supplying meals equal: Cmeals = NF + ML.
  44. 44. 4.8 Optimum location from a welfare viewpoint Total costs C equal Cmeals plus Ctransport , so: tL C = NF + ML + D. 2N
  45. 45. 4.8 Optimum location from a welfare viewpoint dC 2tLD tLD =− 2 + F = 0, so : N = . dN 4N 2F When R = 40, D = 2πR ≈ 251.2, L = 10,000, F = 15,000, M = 15, t = 2, 2 × 10,000 × 251.2 the solution is: ≈ 13 restaurants. 2 × 15,000
  46. 46. 4.8 Optimum location from a welfare viewpoint 800000 700000 600000 500000 400000 300000 200000 100000 0 5 7 9 11 13 15 17 19 21 23 25 27 29 31 C(meals) C(transport) C Figure 4.16: Cost functions
  47. 47. 4.8 Optimum location from a welfare viewpoint 800000 700000 600000 500000 400000 300000 200000 100000 0 5 7 9 11 13 15 17 19 21 23 25 27 29 31 C (meals) C(transport) C TR Figure 4.17: Cost functions and Total Revenue function if the price of a meal equals 34.50.

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