Space and Economics
Chapter 3: Classical Location Theory of the Firm
Author
Wim Heijman (Wageningen, the Netherlands)
July 21, 2009

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

3.1 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

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.

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

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

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

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

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 .

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.

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)

3.3 Trans shipment costs
S O
G M
tg tm
T
Figure 3.6: Location S of a business with transshipment location O

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 .

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 .

3.3 Trans shipment costs
If
K g = Km = Ko.
Then the firm is footloose

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.

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

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.

Alfred Weber (1886 1958)

Pierre Varignon (1654 1722)

3.5 Alfred Weber’s theory on location
C
tc
S
tb
ta
A B
Figure 3.9: The Varignon frame.

3.5 Alfred Weber’s theory on location
Original Varignon Frame

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

3.5 Alfred Weber’s theory on location
+60 -70
V
+40
+20
S
-10
W
Figure 3.10: Transportation cost optimum with isodapanes.

3.5 Alfred Weber’s theory on location
+60 +60
+40 +40
+20 +20
S1 S S2
-60
Figure 3.11: Agglomeration benefits.

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.

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.

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.

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.

www.corusgroup.com/file_source/staticfiles/corus_locations.pdf
3.5 Alfred Weber’s theory on location

3.6 The theory of market areas
Three important authors:
Walter Christaller (1934),
Tord Palander (1935),
August Lösch (1939).

3.6 The theory of market areas

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

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.

3.6 The theory of market areas
Figure 3.19: Seven firms with their market areas.

3.6 The theory of market areas
Figure 3.20: Hexagonal structure and hierarchy of central places.

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

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.

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.

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

Harold Hotelling (1895 1973)

3.9 Application www.sugartech.co.za/factories/index.php
Sugar factory
Figure 3.23: Location of sugar factories in the Netherlands in 1992