Chapter 4: Modern Location Theory of the Firm

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