Italy, Urbino, 20-22 September 2012

7th Workshop "Dynamic Models in Economics and Finance"MDEF 2012

Global dynamics of a three-regionnew economic geography model

Iryna Sushko, Institute of Mathematics, NASU, Ukraine

Pasquale Commendatore, University of Naples ‘Federico II’, Italy

Ingrid Kubin, Vienna University of Economics and Business Administration, Austria

- Introduction

- Description of the 3-region Footloose Entrepreneur New Economic Geography model;

- Peculiarity of the map: coexistence of attractors and Wada basins.

Introduction- The model describes spatial distribution of industrial activity in the long-run across

three identical regions depending on the balancing of agglomeration and dispersion forces.

- It is defined by a 2Dim piecewise smooth continuous noninvertible map depending on

4 parameters.

- Analysis of global dynamics depending on the parameters involves standard bifurcation

theory (Kuznetsov 1995) and border collision bifurcation theory (Nusse and Yorke,

1992, 1995), the theory of critical lines (Mira et al. 1996), characteristic for noninvertible

and piecewise smooth maps, etc.

- Peculiarity of the map is in its symmetry with respect to the main diagonal of the phase

space (any invariant set is either symmetric itself or there exists another invariant set

symmetric to the considered one);

- Coexistence of attractors → basins of attraction?

- Basin of attraction BA of an attractor A is the set of all initial points that are

asymptotic to the attractor (i.e., the points whose ω-limit set belongs to the attractor);

BA consists of an immediate basin (the largest neighborhood of A each point of which is

attracted to A) and all its preimages;

- Basin of attraction can be a simply- or multiply-connected set; it can consists of finite or

infinite (countable or uncountable) number of disjoint sets (Mira et al. 1994, 1996, Bischi

and Lamantia, 2005).

- Basin boundary is an invariant set separating one basin from others (a point belongs to

the boundary if every neighborhood of it intersects different basins).

- The basin boundary can be a smooth curve (e.g., stable manifold of a saddle cycle), or it

can have a fractal structure. A basin boundary is fractal if it contains a transversal

homoclinic point.

- A boundary point is called Wada point if every neighborhood of it has nonempty

intersections with at least three different basins. A basin boundary is called Wada basin

boundary if each of its points is a Wada point (Kennedy and Yorke, 1991).

- Fractal and Wada basin boundaries lead to a final state sensitivity (Grebogi at al. 1993):

for a specific parameter setting and initial condition, no reliable computation can be made

to predict the system’s asymptotic attractor.

- Basin metamorphoses: fractalization, transition from simply- to multiple-connected

basins, etc.

3-Region Footloose Entrepreneur New Economic Geography (NEG)model: Commendatore and Kubin, 2008, 2013

- In each region an agricultural sector (A) and a manufacturing sector (M) are localized.

- Two factors of production: Unskilled labour (L), that does not migrate, is equally

distributed among the regions; Entrepreneurs (N) are mobile across regions.

- The representative consumer’s utility function is:

U = CA1−μ

CMμ

where CA and CM correspond to, respectively, the consumption of the agricultural

(homogeneous) good and of a composite of manufactured goods:

CM = ∑i=1

n

d i

σ−1σ

where d i is the consumption of good i, n is the total number of manufactured goods;

σ > 1 is the constant elasticity of substitution; 1 − μ and μ are, respectively, the invariant

shares of income devoted to the agricultural and manufactured goods, 0 < μ < 1.

- The share of entrepreneurs in region r in period t is

0 ≤ λr,t ≤ 1, ∑r=1

3 λr,t = 1

- N is the total number of entrepreneurs;

- The number of regional varieties produced in period t in region r is

nr,t = λr,tN

- Transportation of the agricultural product between regions is costless.

- Iceberg form of transportation costs for manufactures: if one unit is shipped between

regions r and s, 1/Trs arrives:

Trs = T ≥ 1 for r ≠ s, Trr = 1 otherwise.

- The “trade freeness” parameter is defined as φ ≡ T1−σ.

- With zero transport costs, the agricultural price is the same across the regions.

- Y is the income of the overall economy.

- Total expenditure on the agricultural product is 1 − μY.

- A local price p for its variety is

p = σσ − 1

β

- The effective price in region r for a variety produced in region s is pTrs.

- The regional manufacturing price index facing consumers in region r is given by

Pr,t = ∑s=1

3nsp

1−σTrs1−σ

1

1−σ

- With identical trade costs across regions, we can write

Pr,t = Δr,t

1

1−σ N1

1−σ p

where Δr,t = λr,t + φ1 − λr,t.

- The demand per variety in region r is

dr,t = ∑s=1

3 μYs,tPs,tσ−1Trs

1−σ p−σ = ∑s=1

3ss,tPs,t

σ−1Trs1−σ μYp−σ

where Ys,t represents income and expenditure in region s in period t,ss,t ≡ Ys,t/Y denotes region s’ share in expenditure in period t.

- At a short-run equilibrium for a variety produced in region r: xr,t = dr,t, where xr,t is

the output of each firm located in region r.

- The short-run equilibrium operating profit per variety/entrepreneur in region r is

πr,t = pxr,t − βxr,t =pxr,t

σ- The total profit received by entrepreneurs is μL/σ − μ.

- The short-run equilibrium profit in region r is determined by the spatial distribution of

entrepreneurs and the regional expenditure shares:

πr,t = ∑s=1

3

μYs,tPs,tσ−1Trs

1−σ p1−σ

σ =

= ∑s=1

3

ss,tPs,tσ−1φrs p1−σ μY

σ =μY

σN∑s=1

3ss,t

Δs,tφrs

- With identical trade costs across regions, φrs = φ for r ≠ s, and φrs = 1 for r = s.

- Regional incomes/expenditures are

Yr,t =L3

+ λr,tNπr,t

- Taking into account that λ3, t = 1 − λ1, t − λ2, t, region r’s share in total expenditure

sr,t can be expressed in terms of λ1, t and λ2, t:

sr, t =

σ−μ3

+ λr, tμφss, t

Δs, t+

1−ss, t

Δ3, t

σ − λr, tμ1

Δr, t− φ

Δ3, t

with r = 1, s = 2 or r = 2, s = 1; s3, t = 1 − s1, t − s2, t.

- Given that the agricultural price is 1, the real income of an entrepreneur in region r is:

ωr, tλ1, t,λ2, t = πr, tPr, t

−μ

The central dynamic equation is mimicking the replicator dynamics:

Mr = λr,t 1 + γωr,tλ1, t,λ2, t−∑s=1

3λs,tωs,tλ1, t,λ2, t

∑s=1

3λs,tωs,tλ1, t,λ2, t

Mr indicates the share of entrepreneurs in r without any boundary conditions.

- Taking into account the boundary conditions, the resulting dynamic system is 2Dim

piecewise smooth map Z : λ1, t,λ2, t → Z1λ1, t,λ2, t,Z2λ1, t,λ2, t :

Zrλ1, t,λ2, t =

0 if Mr ≤ 0,

Mr if Mr > 0, Ms > 0, Mr + Ms < 1,

Mr / Mr + Ms if Mr > 0, Ms > 0, Mr + Ms ≥ 1,

Mr / 1 − Ms if Mr > 0, Ms ≤ 0, Mr + Ms < 1,

1 if Mr > 0, Ms ≤ 0, Mr + Ms ≥ 1,

where r = 1, s = 2 for Z1 and r = 2, s = 1 for Z2; Z depends on 4 parameters:

0 < μ,φ < 1, γ > 0 and σ > 1.

Fixed points (Commendatore and Kubin, 2013)

- Core-Periphery (boundary) fix. points: 1,0,0, 0,1,0, 0,0,1;- 2R symmetric fix. points: 0.5,0.5,0, 0.5,0,0.5, 0,0.5,0.5;- 2R asymmetric fix. points: λ1, 1 − λ1, 0, λ1, 0, 1 − λ1 , 0,λ2, 1 − λ2 ;- 3R symmetric interior fix. point: 1/3,1/3,1/3;- 3R asymmetric interior fix, points:

λ1∗,

1−λ1∗

2,

1−λ1∗

2,

1−λ2∗

2,λ2

∗,1−λ2

∗

2,

1−λ3∗

2,

1−λ3∗

2,λ3

∗.

Global dynamics

- 2Dim bifurcation diagram in φ,γ-parameter plane at μ = 0.25, σ = 8 and in

φ,μ-parameter plane at γ = 5, σ = 5

- 2Dim bifurcation diagram in φ,σ-parameter plane at μ = 0.4, γ = 5

- 1Dim bifurcation diagram: φ vs λ1 and φ vs λ2

μ = 0.4, γ = 5, σ = 5, 0.08 < φ < 0.125

The fix. point 1/3,1/3 looses stability with resonance 1 : 2, i.e., both eigenvalues are

equal to −1; Three cycles of period 2 are born.

- 1Dim bifurcation diagram: φ vs λ1 and φ vs λ2

μ = 0.4, γ = 5, σ = 5,0.084 < φ < 0.087

At φ ≈ 0.08625 a fold bifurcation occurs leading to 3 cycles of period 6. The period-2

and period-6 cycles undergo Neimark-Sacker bifurcation leading to closed invariant

attracting curves.

- Coexisting attractors: three 2-cyclic and three 6-cyclic closed invariant curves μ = 0.4,

γ = 5, σ = 5, φ = 0.085

The lines λ1 = 0, λ2 = 0, λ1 + λ2 = 1 (λ3 = 0) are invariant; their preimages are

boundaries of the sets of initial points which are eventually (after finite number of iterations)

belong to these lines. The limit sets on the lines are attractors in Milnor sense.

Wada basins

Contact bifurcation: μ = 0.4, γ = 5, σ = 5

φ = 0.0834 : chaotic attractor after the contact of three 2-cyclic chaotic attractors with

their basin boundaries; φ = 0.0799 : chaotic attractor near disappearence due to a

contact with its basin boundary.

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