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Space and Economics
Chapter 4: Modern Location Theory of the Firm
Author
Wim Heijman (Wageningen, the Netherlands)
July 23, 2009
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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
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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.
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4.1 Neoclassical location theory
Figure 4.1: Location of a firm along a line
L G
0 100
t l
tg
V
T
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4.1 Neoclassical location theory
, MAX 1 αα −= glq
( ) ( ) ( ) ( )( ) . s.t. gtTppltppgtppltppB lgtgl
ltlg
gtgl
ltl −+++=+++=
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4.1 Neoclassical location theory
( )( )
( )( ) .
1
:so ,1
.1000 ,
1 αααα
α
α
−
−+−
+=
−+−=
≤≤+
=
lgtgl
ltl
lgtg
ll
ltl
tTpp
B
tpp
Bq
tTpp
Bg
ttpp
Bl
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4.1 Neoclassical location theory
Assume: ,5.0=α ,100=T ,500=B ,2=lp,5=gp,1.0=ltp .2.0=g
tp
Then:
( )
( ) .501.202.0
500,621002.05
2501.02
250
:so ,1002.05
250 ,
1.02
250
2
5.05.0
++−=
−+
+=
−+=
+=
llll
ll
ttttq
tg
tl
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4.1 Neoclassical location theory
Table 4.1: Inputs and production along a line.
lt l g q
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
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Figure 4.2: Spatial production curve.
4.1 Neoclassical location theory
0
5
10
15
20
25
30
35
40
0 10 20 30 40 50 60 70 80 90 100
Distance from L: tl
Pro
du
ctio
nq
GL
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4.2 The neoclassical optimization problem in a two dimensional space
:and ,),( s.t.
, and ,, respect towith
,)()(Min
*cbac
yxba
ctfbtfpatfpK
ss
ccbbbaaa
=
++++=
,)()(
,)(
,
22
22
22
sccsc
sbsb
ssa
yyxxt
xxyt
xyt
−+−=
−+=
+=
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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.
( ),),(),()()(min *cbacbacftbtfpatfpL ccbbbaaa −−++++= λStep 1:
.//
bbb
aaa
tfp
tfp
bc
ac
++=
∂∂∂∂ ( ).,, cba tttKK =
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4.2 The neoclassical optimization problem in a two dimensional space
Step 2: Because:
,)()(
,)(
,
22
22
22
sccsc
sbsb
ssa
yyxxt
xxyt
xyt
−+−=
−+=
+=
we can now find the optimum with:
,0=∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂=
∂∂
s
c
cs
b
bs
a
as x
t
t
K
x
t
t
K
x
t
t
K
x
K
and:
.0=∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂=
∂∂
s
c
cs
b
bs
a
as y
t
t
K
y
t
t
K
y
t
t
K
y
K
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4.2 The neoclassical optimization problem in a two dimensional space
0 5 10 15 20 25 30 35 40 45 50 55 60
320
310
300
290
280
270
260
250
240
230
220
210
200
190
y=0
y=5
y=10
y=15
y=20
y=25
y=30
y=35
y=40
K
x
Figure 4.3: Spatial costs curves in the neoclassical model.
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4.2 The neoclassical optimization problem in a two dimensional space
K
Figure 4.4: 30D presentation of the neoclassical cost function.
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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
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4.3 Growth poles
Technical growth pole: geographically concentrated supply chain based on forward and backward linkages.
Product Chain
Firm BFirm A Firm C
Backward Linkage Forward Linkage
Semi Finished Product Semi Finished Product
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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.
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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.
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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.
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4.3 Growth poles
TechnicalGrowth Pole
IncomeGrowth Pole
PsychologicalGrowth Pole
Planned
Growth Pole
Figure 4.6: Types of growth poles.
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4.4 Core and periphery
Gunnar Myrdal (189881987): Core periphery
theory:
economic growth inevitably leads to regional
economic disparities.
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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”
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4.4 Core and periphery
� The core regions attract production factors (labour, capital) from the periphery: “backwash8effects”
� If the cumulative causation continues, congestionappears in the core regions (traffic jams, high land prices, high rents, high wages, etcetera).
� This will generate migration of land8intensive and labour8intensive 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”.
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4.4 Core and periphery Alfred Weber’s theory on location
Figure 4.7: The principle of cumulative causation
Improvement ofinfrastructure
Location ofa pull element
Expansion of
goods and servicesfor the local market
Increase of localtax revenues
Psychologicalpolarisation
Technicalpolarisation
Growth ofemployment andincome:income polarisation
production of
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Gunnar Myrdal (189801987)
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4.5 Agglomeration and externalities
Figure 3.12: Spatial margins to profitability.
� Economies of scale: costs per unit product decrease if the scale of production increases
� Two types of externalities:
8 internal;
8 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 clusteror agglomeration
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4.5 Agglomeration and externalities
, if ,0 , if ,0 , if ,0
,0, ),(
***ss
s
sss
s
sss
s
s
sssss
NNdN
dKNN
dN
dKNN
dN
dK
NKNKK
==>><<
≥=
0.,, ,2 >+−= γβαγβα sss NNK
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4.5 Agglomeration and externalities
Figure 4.8: Stable spatial equilibrium.
K1K2
1 2N1 N2
N
O A B C
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4.5 Agglomeration and externalities
Figure 4.9: Unstable spatial equilibrium.
K1
K2
N1 N2
O A B C
N
1 2
D
E
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4.5 Agglomeration and externalities
.2
:so ,02
,
*
2
αββα
γβα
==−=
+−=
sss
s
sss
NNdN
dK
NNK
.2
**
βαN
N
Nm
s
==
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4.5 Agglomeration and externalities
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4.5 Agglomeration and externalities
http://www.liof.com/?id=28
www.emcc.eurofound.eu.int/automotivemap
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4.6 Market forms: spatial monopoly
Figure 4.12: Spatial demand curve.
α−−= ||)( sxxKxq , Txx ≤≤0 10 << α
MSPxs
xT0 x
q(x)
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4.6 Market forms: spatial monopoly
.)()()(0
dxxxKdxxxKxQT
s
s x
x
s
x
s ∫∫−− −+−= αα
.2T
s
xx =
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4.7 Spatial duopoly: Hotelling’s Law generalised
MSP
x
xT0
q (x)
2MSP1
1
q (x)2
q (x)2
q (x)1
xx1 20.5( + )x x1 2
q (x)1
q (x)2
Figure 4.13: Spatial duopoly with two mobile selling points (MSP).
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4.7 Spatial duopoly: Hotelling’s Law generalised
∫ ∫
∫∫
+
−−
+
−−
−+−=
−+−=
2
212
21
1
1
)(2
1222
)(2
1
1
0
11
.)()()(
,)()()(
x
xx
x
x
xx
x
x
T
dxxxKdxxxKxQ
dxxxKdxxxKxQ
αα
αα
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4.7 Spatial duopoly: Hotelling’s Law generalised
The cooperative solution :
.43
,41
21 TT xxxx ==
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4.7 Spatial duopoly: Hotelling’s Law generalised
competitive solution:
TT
T
xxxx
xx
21
21
22
,21
21
2
1
1
12
1
1
1
+
+=−=
+=
α
α
α
α
The competitive solution represents a so0called Nash equilibrium.
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4.7 Spatial duopoly: Hotelling’s Law generalised
If ,∞→α then Txx41
1 → and ,43
2 Txx → which is equal to the cooperative
(efficient) solution.
If ,0→α then ,21
, 21 Txxx → which is the Hotelling Law (Section 3.7).
For ,0 ∞<< α ,21
41
1 TT xxx << and .43
21
2 TT xxx <<
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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.
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4.8 Optimum location from a welfare viewpoint
Figure 4.14: Six restaurants in a circular space.
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4.8 Optimum location from a welfare viewpoint
DN
d1
21=
The cost per unit distance equals t, so the total transportation costs transportC for L
customers equal:
.2transport D
N
tLC =
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4.8 Optimum location from a welfare viewpoint
With constant marginal costs M and fixed costs per restaurant F, and Q meals, the costs mealsC of the meals are:
.meals MQNFC += If there is one meal per customer per day, then, with L customers and N restaurants, total costs per day mealsC for supplying meals equal:
.meals MLNFC +=
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4.8 Optimum location from a welfare viewpoint
Total costs C equal mealsC plus ,transportC so:
.2
DN
tLMLNFC ++=
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4.8 Optimum location from a welfare viewpoint
.2
:so ,042
2 F
tLDNF
N
tLD
dN
dC ==+−=
When ,40=R ,2.2512 ≈= RD π ,000,10=L ,000,15=F ,15=M ,2=t
the solution is: 13000,152
2.251000,102 ≈×
×× restaurants.
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4.8 Optimum location from a welfare viewpoint
Figure 4.16: Cost functions
0
100000
200000
300000
400000
500000
600000
700000
800000
5 7 9 11 13 15 17 19 21 23 25 27 29 31
C(meals) C(transport) C
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4.8 Optimum location from a welfare viewpoint
Figure 4.17: Cost functions and Total Revenue function if the price of a meal equals € 34.50.
0
100000
200000
300000
400000
500000
600000
700000
800000
5 7 9 11 13 15 17 19 21 23 25 27 29 31
C (meals) C(transport) C TR