chapter 4: modern location theory of the firm
TRANSCRIPT
Space and Economics
Chapter 4: Modern Location Theory of the Firm
Author
Wim Heijman (Wageningen, the Netherlands)
July 23, 2009
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
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.
4.1 Neoclassical location theory
Figure 4.1: Location of a firm along a line
L G
0 100
t l
tg
V
T
4.1 Neoclassical location theory
, MAX 1 αα −= glq
( ) ( ) ( ) ( )( ) . s.t. gtTppltppgtppltppB lgtgl
ltlg
gtgl
ltl −+++=+++=
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
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
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
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
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
−+−=
−+=
+=
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 =
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
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.
4.2 The neoclassical optimization problem in a two dimensional space
K
Figure 4.4: 30D presentation of the neoclassical cost function.
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
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
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.
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.
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.
4.3 Growth poles
TechnicalGrowth Pole
IncomeGrowth Pole
PsychologicalGrowth Pole
Planned
Growth Pole
Figure 4.6: Types of growth poles.
4.4 Core and periphery
Gunnar Myrdal (189881987): Core periphery
theory:
economic growth inevitably leads to regional
economic disparities.
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”
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”.
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
Gunnar Myrdal (189801987)
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
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
4.5 Agglomeration and externalities
Figure 4.8: Stable spatial equilibrium.
K1K2
1 2N1 N2
N
O A B C
4.5 Agglomeration and externalities
Figure 4.9: Unstable spatial equilibrium.
K1
K2
N1 N2
O A B C
N
1 2
D
E
4.5 Agglomeration and externalities
.2
:so ,02
,
*
2
αββα
γβα
==−=
+−=
sss
s
sss
NNdN
dK
NNK
.2
**
βαN
N
Nm
s
==
4.5 Agglomeration and externalities
4.5 Agglomeration and externalities
http://www.liof.com/?id=28
www.emcc.eurofound.eu.int/automotivemap
4.6 Market forms: spatial monopoly
Figure 4.12: Spatial demand curve.
α−−= ||)( sxxKxq , Txx ≤≤0 10 << α
MSPxs
xT0 x
q(x)
4.6 Market forms: spatial monopoly
.)()()(0
dxxxKdxxxKxQT
s
s x
x
s
x
s ∫∫−− −+−= αα
.2T
s
xx =
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).
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
4.7 Spatial duopoly: Hotelling’s Law generalised
The cooperative solution :
.43
,41
21 TT xxxx ==
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.
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 <<
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.
4.8 Optimum location from a welfare viewpoint
Figure 4.14: Six restaurants in a circular space.
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 =
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 +=
4.8 Optimum location from a welfare viewpoint
Total costs C equal mealsC plus ,transportC so:
.2
DN
tLMLNFC ++=
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.
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
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