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2018 | OriginalPaper | Buchkapitel

5. High-Dimensional Nonlinear Cobweb Model

verfasst von : Tamotsu Onozaki

Erschienen in: Nonlinearity, Bounded Rationality, and Heterogeneity

Verlag: Springer Japan

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Abstract

One of the important conclusions of Chap. 3 is that heterogeneity matters decisively in the complex behavior of a nonlinear economy when at least two different types of agents exist. The question then arises: what happens if there are many different agents in a nonlinear economy? This chapter investigates such a problem by concentrating on synchronization among producers’ chaotic behavior. For the sake of simplicity, behavioral heterogeneity is ignored and producers are considered to be identical in the model; however, they can be deemed to be heterogeneous in the sense that the initial conditions of producers are randomly selected. Recall that owing to sensitive dependence on initial conditions, a chaotic map can generate completely different orbits for different initial conditions.

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Vorheriges Kapitel From Nonlinear Economic Dynamics to Complexity Economics

Nächstes Kapitel Agent-Based Model of Market Structure Dynamics I

The content of this chapter is mainly based upon Onozaki et al. (2007) and Esashi et al. (2017).

Synchronization has also received much attention in finance in the 21st century. For a survey of the related literature, see Huang and Chen (2015).

In this way, in coupled oscillator systems, whether the interaction among oscillators is direct or indirect, separate oscillators may synchronize via entrainment.

Coupled map lattice (CML), a framework similar to the GCM model, was also first proposed by Kaneko (1992). The main difference between GCM and CML is that the former include the mechanism of the global interaction of all elements, whereas the latter include that of the local interaction. For an application of the CML system in finance, see Huang and Chen (2015).

A many-body system with interactions, often considered in physics and probability theory, is generally difficult to solve exactly. Therefore, the many-body system is replaced by a one-body system with an appropriate external field, which is considered to be an approximation of the effect of all the other individuals on a given individual. This external field, usually represented by a single averaged effect, is called a mean field.

This is sometimes discussed from the viewpoint of the Milnor attractor (Milnor (1985), Kaneko (2002)).

We neglect interregional trade in the first step, since by considering it, we would have to assume a new economic agent (i.e., a distributor), whose behavioral objective is different from that of a producer. Such an extension would make the model overly complicated and intractable.

By applying a variable transformation of \(x(i)=n^{-\frac{\eta }{1+\eta }}z(i)\), we can cancel out the parameter N from the model. As the transformation is linear, the qualitative behavior of x(i) remains perfectly preserved. The variable x(i) appearing in the following is read as z(i).

In statistical mechanics, the ensemble average is defined as the arithmetic mean of a quantity that is a function of the individual states of the system. Here the duration of the synchronous state is regarded as a function of the individual outcome of the system with respect to the different initial conditions.

This index is defined as the sum of the squares of the market shares of the relevant individual firms in economics. In physics, the same index was first introduced in the spin glass Mézard et al. (1987) and is applied to complex chaotic dynamics Kaneko (1991).

The index H ranges from 1/C to 1, while the index \(H^*\) ranges from 0 to 1. When \(C=1\), it is assumed that \(H^*=0\) to avoid division by zero.

The value of \(\delta \) is fixed at \(10^{-4}\) in the remainder of the chapter.

See Sect. 7.2.4 for more detailed explanation of power law. Although this kind of power law distribution is observed in a well-known critical phenomenon called type III intermittency in low-dimensional systems, it is not robust in the parameter space. By contrast, the distributions in our model are robustly found.

The value of \(\tau \) is fixed at \(10^{5}\) in the remainder of the chapter.

The MEDs are calculated from 10 randomly chosen initial conditions \(x_{0}(i)~(i=1,\dots , 10)\) after neglecting the transient of \(10^5\) iterations. If the average is a noninteger, the corresponding set of parameters \((\phi , \varepsilon )\) is plotted in Fig. 5.10. These calculations are performed for the \(10^5\) points in a parameter region \((\phi ,\varepsilon )\).

Although we do not change the parameter value of \(\eta \), we confirm the numerical evidence that chaotic itinerancy is observed robustly in a wide range of parameter constellations of \((\varepsilon , \phi )\). Since the parameter \(\eta =3.5\) is not a singular value for which the system has special symmetry, chaotic itinerancy should appear when making even a slight change in \(\eta \) from 3.5.

This is identical to the 10-dimensional map (\(N=10\)), where the initial conditions are given as \(x_{0}(1)=x_{0}(3)=x_{0}(5)=x_{0}(7)=x_{0}(9)\) and \(x_{0}(2)=x_{0}(4)=x_{0}(6)=x_{0}(8)=x_{0}(10)\). If \((x_{t}(1), x_{t}(2))\) is a solution, \((x_{t}(2), x_{t}(1))\) is also a solution because of the symmetry within the system.

Homoclinic tangency and unstable dimension variability are considered to be typical structures that break hyperbolicity Bonatti et al. (2005).

It is left for the readers to confirm that the fixed point \((x(1), x(2))=(1,1)\) is a repellor, namely whether the two eigenvalues of the Jacobian matrix of (5.9) evaluated at the fixed point \((x(1), x(2))=(1,1)\) are greater than unity in absolute value, under the constellation of the parameters \((\eta , \phi , \varepsilon )=(3.5, 0.7, 0.315)\) as assumed in Sect. 5.4.

We empirically investigate multi-regional business cycle synchronization and find that business cycles in more regions tend to behave similarly in the recession stage, whereas those in fewer regions do so in the recovery/expansion stage. This result is obtained by applying cross-wavelet analysis to monthly data (from January 1977 to December 2011) on Japan’s index of industrial production by prefecture (original index, 2005 average \(=\) 100) for all of 47 prefectures. We are now preparing a separate paper.

Titel: High-Dimensional Nonlinear Cobweb Model
verfasst von: Tamotsu Onozaki
Verlag: Springer Japan
Buch: Nonlinearity, Bounded Rationality, and Heterogeneity
Print ISBN: 978-4-431-54970-3

Electronic ISBN: 978-4-431-54971-0

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-4-431-54971-0_5