Abstract
This chapter deals with the following basic questions from the historical and methodological perspectives: what is nonlinear economic dynamics and how has it been developed? After introducing some of the basic concepts of nonlinear economic dynamics in Sect. 1.1, we discuss how economic dynamics was formed in the 1930s and the extent to which it is closely related to the equilibrium paradigm, which is synonymous with the neoclassical paradigm, in Sect. 1.2. In Sect. 1.3, by comparing two approaches in economic dynamics, namely exogenous and endogenous business cycle theories, we state that the exogenous business cycle theory relied on linear dynamical models and was suitable for the equilibrium paradigm, while the endogenous business cycle theory relied on nonlinear dynamical models and was the starting point of nonlinear economic dynamics.
If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things.
René Descartes Principles of Philosophy (1644)
Then you will know the truth, and the truth will set you free.
Bible (John 8:32)
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- 1.
This book only considers time-invariant systems because it is difficult to deal with time-variant systems even if they are linear.
- 2.
Throughout this book, we use this notation only for functions to express their iterations. Readers should be cautious not to confuse this with the power of functions.
- 3.
This concept of an equilibrium coincides with that of a fixed point defined in Sect. 1.1.
- 4.
In continuous dynamical system notation, (1.6) is rewritten as
$$\begin{aligned} \dot{p^{i}} = f^{i} ( d^{i}(p) - s^{i}(p) ), \quad i=1, 2, \ldots , n, \end{aligned}$$and the dynamical equilibrium is defined as \(\dot{p^{i}}=0, i=1, \ldots , n\).
- 5.
Schumpeter (1954) states, “always static theory has historically proceeded dynamical theory and the reasons for this seem to be as obvious as they are sound—static theory is much simpler to work out; its propositions are easier to prove; and it seems closer to (logical) essentials (p. 964).” The last point, namely whether static theory seems closer to essentials, is highly dependent on what we analyze and is not approved. Rather, from the viewpoint of nonlinear economic dynamics, it may be much more important to explain that a system has dynamical properties that cannot be explained by static theory.
- 6.
The notion of asymptotical stability is defined in Sect. 2.1.
- 7.
Because we restrict ourselves to discrete-time models in this book, we do not give an account of the limit cycles that appear in continuous-time models. These details are left to other books, such as Guckenheimer and Holmes (1983) and Lorenz (1993). Note that the limit cycles in continuous models correspond to the periodic cycles in discrete models.
- 8.
We deal with Laplace’s demon in Sect. 1.5.
- 9.
As regards the relation between physics and economics, see, e.g. Sebba (1953), Georgescu-Roegen (1971), Jojima (1985), Mirowski (1988), Ingrao and Israel (1987), and Lorenz (1993).
- 10.
For detailed discussions on this methodological attitude, see West (1985) and Lorenz (1993).
- 11.
We criticize the framework of neoclassical microeconomics from a different viewpoint in Sect. 6.1.
- 12.
Cited from Ingrao and Israel (1987), p. 159. For the original French text, see Jaffé (1965), vol. 3, p. 163, letter 1496.
- 13.
The description of what would later be called bounded rationality first appeared as the “limits of rationality” in Simon (1947). The term bounded rationality first appeared in Simon (1957).
- 14.
The term “satisficing”, a composite of “satisfy” and “suffice,” was coined by Simon (1956).
- 15.
Sargent (1993) states, “(w)e can interpret the idea of bounded rationality broadly as a research program to build models populated by agents who behave like working economists or econometricians.” (p. 22).
- 16.
See Mäki (2009).
- 17.
We introduce the notions of attractor and homoclinic orbit in Chap. 2. For further details about these notions, see, for example, Palis and Takens (1993).
- 18.
Although Lorenz is often referred to as the first discoverer of a strange attractor (the Lorenz attractor), Ueda (a graduate student at that time) actually discovered a strange attractor (the Ueda attractor) a little earlier. See Abraham and Ueda (2001) for details.
- 19.
The Li–Yorke theorem is introduced in the next chapter. For dramatic stories of founding “chaos theory”, see Gleick (1987). A tutorial survey of chaos theory is left to introductory books on this subject.
- 20.
For a survey of applications of chaos theory to economics, see Goodwin (1990), Chiarella (1990), Medio (1992), Lorenz (1993), Day (1994, 1999), Puu (2000), Rosser (2000), Medio and Lines (2001), Zhang (2006), Bischi et al. (2009), Bischi et al. (2010), Bischi et al. (2013), and Hommes (2013). For neoclassical nonlinear equilibrium dynamics, see Grandmont (1987), Benhabib (1992), and Majumdar et al. (2000).
- 21.
The existence of these works by Palander is discussed in Puu (2002).
- 22.
On such methods, see Abarbanel (1996) and Kantz and Schreiber (1997). For applications to economic data, see Lorenz (1993).
- 23.
For the BDS test, see Brock et al. (1991).
- 24.
For a proof of the theorem, see Sharkovsky (1964), Guckenheimer and Holmes (1983), Block and Coppel (1992), or Elaydi (2007).
- 25.
To be precise, the chaotic behavior in this case is called topological chaos. As discussed in the next chapter, a map exhibits topological chaos either if it has a horseshoe or, alternatively, if the topological entropy of the map is positive.
- 26.
It is easy to demonstrate that high-dimensional nonlinear systems can generate any autocorrelation structure. See Footnote 6 in Hommes (2001), p. 152.
- 27.
We will introduce the notion of repellors in Chap. 2.
- 28.
For the details of the Newhouse phenomenon, see Newhouse (1974) and Palis and Takens (1993).
- 29.
- 30.
Crisis-induced intermittency was first found by Grebogi et al. (1983). We introduce the notion of crisis in Chap. 2. On-off intermittency was first found by Fujisaka and Yamada (1983) and Yamada and Fujisaka (1983). Regarding economics, Bischi et al. (1998), Bischi and Gardini (2000), and Huang and Chen (2014) study the synchronization of chaotic oscillators in two-dimensional models, all of which exhibit on-off intermittency.
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Onozaki, T. (2018). The Nature and Significance of Nonlinear Economic Dynamics. In: Nonlinearity, Bounded Rationality, and Heterogeneity. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54971-0_1
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