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Über dieses Buch

This book pursues a nonlinear approach in considering both chaotic dynamical models and agent-based simulation models of economics, as well as their dynamical behaviors. Three key concepts arising in this context are “nonlinearity,” “bounded rationality” and “heterogeneity,” which also make up the title of the book. Nonlinearity is the warp that runs throughout all models because systems that exhibit chaotic or other complex behavior in the absence of any exogenous disturbances are absolutely nonlinear. Bounded rationality constitutes the woof, because economic systems do not exhibit complex behavior if all agents are perfectly rational, as is usually assumed in neoclassical economics. Agents who are boundedly rational have to struggle to do their best with limited information and tend to adapt to their economic environment without knowing what is the best. Furthermore, the heterogeneity of firms or consumers dyes the fabric of complex dynamics woven from the warp and woof.



Erratum to: One-Dimensional Nonlinear Cobweb Model

Tamotsu Onozaki

Nonlinear Economic Dynamics


Chapter 1. The Nature and Significance of Nonlinear Economic Dynamics

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.
Tamotsu Onozaki

Chapter 2. One-Dimensional Nonlinear Cobweb Model

This chapter provides an introductory exposition of nonlinear discrete dynamics, which takes readers into the vast ocean outside the equilibrium paradigm, by presenting a one-dimensional nonlinear cobweb model. The model is a simple extension of the standard cobweb model. The new ingredients are an iso-elastic inverse demand function (i.e., an inverse demand function with constant price elasticity) and boundedly rational producers who gradually adjust their production toward the target levels based upon naive price expectations. Thus, the key parameters of the model are the price elasticity of demand and the production adjustment speed of producers. With the aid of mathematical analysis and numerical simulations, it is shown that for a large set of parameter values, the cobweb market exhibits observable chaos (a strange attractor) as well as topological chaos (a horseshoe) associated with homoclinic points.
Tamotsu Onozaki

Chapter 3. Two-Dimensional Nonlinear Cobweb Model

This chapter extends the one-dimensional model studied in Chap. 2 to include two different types of producers in order to investigate whether slight behavioral heterogeneity could be a factor that drastically changes the dynamical properties of a market. The two types of producers are naive optimizers and cautious adapters. A naive optimizer produces the profit-maximizing quantity instantaneously, while a cautious adapter adjusts his/her output toward the profit-maximizing quantity as a target. We obtain a two-dimensional model, which is more difficult to analyze because mathematical theories of higher-dimensional nonlinear dynamical systems are underdeveloped compared with those of one-dimensional systems. With the aid of mathematical analysis as well as numerical calculations, we show that a single agent may change the complexity of market behavior. In a market of naive optimizers, a single cautious adapter stabilizes the otherwise exploding market. In a market of cautious adapters, a single naive optimizer may destabilize the market. Without him/her, there exists at most one periodic attractor in the market. However, with him/her, many (and even infinitely many) coexisting periodic attractors may appear.
Tamotsu Onozaki

Toward Economic Complexity


Chapter 4. From Nonlinear Economic Dynamics to Complexity Economics

In Part I, we discussed the significance of nonlinear economic dynamics and investigated two simple models that exhibit chaotic behavior. Nonlinear economic dynamics originated in the 1930s, led into chaotic economic dynamics at the end of the 1970s, and continues today. However, research on nonlinear economic dynamics has thus far suffered from the serious restriction on mathematical analytics. We discuss this restriction in Sect. 4.1. In Sect. 4.2, we consider where nonlinear economic dynamics should be headed and state that it should aim to use computationally oriented research methods against the background of complex system theory. This statement underpins the studies presented in Chaps. 5, 6, and 7. Furthermore, Sect. 4.3 points out that two fundamental directions exist in which research on economic complexity has been carried out: the econophysics approach and agent-based model approach. We concentrate on the latter approach in Chaps. 6 and 7. This short chapter also serves as a manifestation of our methodological standpoint in Part II.
Tamotsu Onozaki

Chapter 5. High-Dimensional Nonlinear Cobweb Model

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.
Tamotsu Onozaki

Chapter 6. Agent-Based Model of Market Structure Dynamics I

In the remaining two chapters, we investigate the time evolution of the market structure in terms of market share dynamics, by employing agent-based models (ABMs) with boundedly rational firms and consumers interacting with each other. The common basic feature of boundedly rational firms is that since they do not know the shape of their demand functions they face, they adaptively revise production levels and prices so as to raise their profits based on the reaction by consumers. The main difference lies in consumers’ behavior.
Tamotsu Onozaki

Chapter 7. Agent-Based Model of Market Structure Dynamics II

In this final chapter of the book, we continue to investigate a competitive market as a complex adaptive system consisting of mutually interacting, boundedly rational firms and consumers. In the model presented in the previous chapter, product differentiation was considered, showing that a consumer’s brand loyalty plays an important role in the emergence of oligopoly. By contrast, in this chapter, we consider a market of homogeneous goods. In this market, boundedly rational consumers decide from which firm to purchase goods to increase their utility, and we employ, as a first step, a statistical description to represent firms’ distribution of consumer share because the number of consumers is very large. Firms’ distribution is characterized by a single parameter representing how “rationally” the mass of consumers pursue higher utility. Aggregate consumer behavior is described by the Boltzmann distribution, which indicates how rationally the consumer seeks to increase his/her utility. Since the boundedly rational firms do not know the shape of the demand function they face, they adaptively revise their production levels and prices to raise their profits with the aid of a simple reinforcement learning algorithm (i.e., by learning through experience).
Tamotsu Onozaki


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