This document discusses modern location theory of the firm. It covers several topics: - Neoclassical location theory which assumes substitutability of production inputs using a Cobb-Douglas production function. - A two dimensional optimization problem to determine optimal location and input levels. - The concept of growth poles which are geographic concentrations of economic activity that can stimulate regional growth. - Core-periphery theory which describes how economic growth becomes concentrated in core regions. - Agglomeration and externalities and how spatial equilibrium can be stable or unstable based on these factors. - Spatial monopoly and duopoly models and how firm locations are determined under different market structures. - Optimal location from a

1. Space and Economics
Chapter 4: Modern Location Theory of the Firm
Author
Wim Heijman (Wageningen, the Netherlands)
July 23, 2009

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. 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. 4.1 Neoclassical location theory
T
t
l
tg
0 100
L G
V
Figure 4.1: Location of a firm along a line

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. 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. 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. 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. 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. 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. 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. 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. 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. 4.2 The neoclassical optimization problem in a two dimensional space
K
Figure 4.4: 3 D presentation of the neoclassical cost function.

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. 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. 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. 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. 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. 4.3 Growth poles
Technical
Growth Pole
Psychological Income
Growth Pole Growth Pole
Planned
Growth Pole
Figure 4.6: Types of growth poles.

21. 4.4 Core and periphery
Gunnar Myrdal (1898 1987): Core periphery
theory:
economic growth inevitably leads to regional
economic disparities.

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. 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. 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. Gunnar Myrdal (1898 1987)

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. 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. 4.5 Agglomeration and externalities
K2 K1
O A B C 2
1
N1 N2
N
Figure 4.8: Stable spatial equilibrium.

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. 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. 4.5 Agglomeration and externalities

32. 4.5 Agglomeration and externalities
http://www.liof.com/?id=28
www.emcc.eurofound.eu.int/automotivemap

34. 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.

35. 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

36. 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).

37. 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

38. 4.7 Spatial duopoly: Hotelling’s Law generalised
The cooperative solution :
1 3
x1 = xT , x2 = xT .
4 4

39. 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.

40. 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

41. 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.

42. 4.8 Optimum location from a welfare viewpoint
Figure 4.14: Six restaurants in a circular space.

43. 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

44. 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.

45. 4.8 Optimum location from a welfare viewpoint
Total costs C equal Cmeals plus Ctransport , so:
tL
C = NF + ML + D.
2N

46. 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

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
Figure 4.16: Cost functions

48. 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.