Chapter 4: solutions, simulations and extensions

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Chapter 4solutions, simulations and extensions

• Analytical lt-gt numerical
• Long term behaviour does not allow analytical
  solution
• Solution depends on
• Regional distribution of A and M at the start
  (history)
• f1, f 1, , f n
• ?1, ?1, , ?n
• The key parameters d, e and T (economics)
• The distance matrix D (geography)

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Numerical solution

• Yr d ?r Wr fr ( 1 d)
• Ir ( Ss ?s Trs1-e Ws1-e )1/(1-e)
• Wr ( Ss Ys Trs1-e Ise-1 )1/e
• wr Wr Ir-d
• 5. d?1/?1 ?(w1 ?)

• For two regions
• Set W1 W2 1
• Calculate Y and I from 1-2
• Calculate W from 3
• Recalculate Y, I and W
• Stop when for all regions r Wr /Wr lt 1 s
  (break-off condition)
• short term equilibrium
• Calculate wr and ?r from 4-5
• Stop when for all regions r wr /? lt 1 s
  (break-off condition)
• Long term term equilibrium

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Simulation example for two regions
Table 4.1 Base
-
scenario parameter configuration, 2 regions

d
g
0.4

0.4

L

1





r
0.8

b
0.8

f
f
0.5








1
2
a
s
T

1.7

0.08

0.0001





e 1/(1-?) 5
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Result of 59 simulations of the two region model
with varying values of ?1 ?2 1 ?1 using the
parameters of table 4.1

Figure 4.1 The relative real wage in region 1
T 1.7

ws Ws Is-d
stable
unstable
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Figure 4.2 The impact of transport costs


Higher T spreading more likely
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The impact of some parameters

Higher ? spreading more likely Lower ? less
substitution possible -gt nr of varieties more
important -gt agglomeration advantage
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The impact of some parameters
Low d spreading more likely High d M-goods more
important in the budget -gt agglomeration
advantage for local welfare
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Fig 4.3 The Tomahawk diagram
S1 (T1,81)
1
?1
B (T1,63)
0,5
0
S0 (T1,81)
T
Unstable equilibria
Stable equilibria
Break point point where the spreading
equilibrium changes from stable to unstable
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Result of 59 simulations of the two region model
with varying values of ?1 ?2 1 ?1 using the
parameters of table 4.1

Figure 4.1 The relative real wage in region 1
T 1.7

ws Ws Is-d
stable
unstable
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No black hole condition and break point

• Suppose all manufacturing agglomerated into
  region 1
• Solution W11, Y1(1d)/2, Y2(1-d)/2, I11,
  I2T, w11
• Workers will only move to region 2 when w2 gt 1
• w2e f(T)(1d)/2T-(?d)e (1- d)/2T(?-d)e gt
  1
• With increasing T first term becomes small but
  second term only becomes large when ?gtd
• Define g(T) 1-T1-e /1 T1-e 1 -
  d(1?)/(d2?)
• B Tbreak when g(T)1

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w2
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Simulations with a race track economy (4.5.6)

Figure 4.12 The racetrack economy number of
locations R

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Figure 4.12 continued
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Figure 4.10 continued