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A 3-region new economic geography model in discrete time

Pasquale CommendatoreIngrid Kubin

Iryna Sushko

NEG - Central question

• Long-run spatial distribution of industry

2

equally distributed

agglomerated in one region

unevenly distributed

3

NEG – Basic structure

• Given amount of productive factors • Distributed across two identical regions• Allowed to move freely between regions

according to factor rewards (dynamic law)• Output sold in home region as well as in other

regions (trade cost)• Decisive for location decision: Cost for

commodity trade between regions

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• Where do dynamic processes play a role?- Goods and labour markets: istantaneous

equilibrium- Shipping of goods: istantaneous- Factor mobility: gradual over time, adaptive

process- Analytical core: One-dimensional differential

equation

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Continuous Vs discrete time

• 2-R NEG in continuous time (1-D bimodal flow):Multiple equilibria; catastrophic agglomeration; hysteresis; ‘regular’ local stability properties which holds within well-defined basins of attraction• 2-R NEG in discrete time (1-D bimodal map): Multiple equilibria; catastrophic agglomeration; hysteresis; multiple attractors of any periodicity; chaotic dynamics; agglomeration via volatility

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Continuous Vs discrete time

• NEG models in discrete time:CP (Currie&Kubin; JEBO, 2006) [CK, 2006]FE (Commendatore, Currie&Kubin; SEA, 2008) [CCK, 2008]FC (Commendatore, Currie&Kubin; NDPLS, 2007) [CCK, 2007]FE 3-R (Commendatore&Kubin, 2012) – Local stability analysisFE 3-R (Commendatore, Kubin & Sushko, current) – Global stability analysis

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Footloose Entrepreneur (FE)(Forslid & Ottaviano, 2003, JEG)

• Factors of production - unskilled workers (immobile, variable costs) - entrepreneurs (mobile, fixed cost)• Agglomeration and dispersion forces are all at

work• Self-reinforcing agglomeration processes

preserved• Entrepreneurs migrate in response to

differences in real profits

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3-R Footloose Entrepreneur model

• 3 symmetric Regions (1, 2, 3)• 2 Sectors (agriculture,

manufacturing)• 2 Factors of production (unskilled

workers, entrepreneurs)

Manufacturing sector

• Differentiated goods• Produced with:

1 entrepreneur units of unskilled labour

• Decreasing average costs• Monopolistic competition:• Price: mark-up over marginal

cost

• Iceberg transport cost, T

Agricultural sector

• Undifferentiated good• Produced with:

1 unit of unskilled labour

• Constant average costs• Perfect competition • MR = MC

• No transport cost (perfect trade freeness)

1Ap w

1

p w

9

10

12 21T T T 12 13 23T T T T

1

1

TTrade freeness

3 equidistant regions

1 2

3

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• Number of firms = number of entrepreneurs N• number of firms located in region r:

• Share of entrepreneurs (firms) located in region r:

, ,r t r tn N

,r t

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Istantaneous Short-run equilibriumin region r

Operating profit:

Price index:

σ : CES – taste for variety

13 1

1 1, ,

1r t s t rs

s

P n p T

regional expenditure share

1 1

, ,, 1

, , ,

R

s t s t rsr t s

r t r t r t

p Y P T ppx

px x

µ : share of expenditure allocated to manufacturing

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Real profit in region r

,

, 1, 2, 3,,

( , , ) r tr t t t t

r tP

Operating profit per variety in region r

Price index in region r

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Central dynamic equation: non-linear, two-dimensional in

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1, 2,,t t

3

, , ,1

, 3

, ,1

1r t s t s t

sr r t

s t s ts

M

Replicator dynamics

𝜆1, 𝑡+𝜆2 ,𝑡+𝜆3 ,𝑡=1𝑎𝑛𝑑0≤ 𝜆𝑟 , 𝑡≤1but bounduary conditions

: migration speed

1515

, 1 1, 2,

0 0

0, 0, 1

0, 0, 1,

0, 0, 11

1 0, 0, 1

r

r r s r s

rr s r s

r sr t t t

rr s r s

s

r s r s

if M

M if M M M M

Mif M M M M

M MZ

Mif M M M M

M

if M M M M

Central dynamic equation: non-linear, two-dimensional in 1, 2,,t t

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𝜆1∗

𝜆2∗

0 0.5 1

0.5

1

Core-Periphery equilibria

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𝜆1∗

𝜆2∗

0 0.5 1

0.5

1

3-Region symmetric equilibrium

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𝜆1∗

𝜆2∗

0 0.5 1

0.5

1

3-Region asymmetric equilibria

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𝜆1∗

𝜆2∗

0 0.5 1

0.5

1

2-Region ‘symmetric’ equilibria

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𝜆1∗

𝜆2∗

0 0.5 1

0.5

1

2-R asymmetric equilibria (

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3-Region Tomahawk diagrams

131

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Local stability results

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Global dynamics preliminary results

𝜇=0.45 ,𝛾=10 ,𝜎=2.5 ,𝜙=0.275

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Global dynamics preliminary results

𝜇=0.4 ,𝛾=5 ,𝜎=5 ,𝜙=0.085

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Global dynamics preliminary results

𝜇=0.4 ,𝛾=5 ,𝜎=5 ,𝜙=0.085

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Final remarks

• 2-Dimensional nonlinear map: Multiple equilibria; catastrophic agglomeration; hysteresis; 2-R and 3-R stationary equilibria, possible 2-R asymmetric stable equilibriaperiodic and quasi-periodic multiple attractors; chaotic dynamics; strange attractors, complex basins of attraction

• Simple extensions: slight increase in the number of regions; asymmetric trade costs

• Less simple extensions: larger increase in the number of regions, endogenous trasport costs, network structure