SALVIATI: Yesterday took us into such good digressions on Prof. Feynman's words and his van that I do not know whether I shall be able to go ahead without your assistance in putting me back on the track of macro-econophysics.

SAGREDO: You do not need to worry, Salviati. Together with Simplicio, I read this book to my pleasure. I found it to contain many beautiful considerations which are novel and impressive. … You shake your head, Simplicio and smile as if I uttered some absurdity.

SIMP.: I merely smile, but believe me. I have an impression that all matters of economy are not covered and the authors do not have answers on business cycles, not to mention how to predict an economic crisis and how to deal with it.

SALV.: I may remind you that we do not yet have a basic understanding of the underlying dynamics of anything at the micro-level: Anyone can write out models of interactions between economic agents like firms and financial institutions, and simulate the dynamics on simulated networks. But there are so many parameters and so many artificial networks.

SAGR.: Ahuh… They have real credit and trade networks, don't they, which can serve as a basis for all the new directions of work?

SALV.: Doubtless there are. We need all the analytical and mathematical tools and modeling ideas exposed in this book to strive for a real understanding of the nature of economic dynamics.

SIMP.: You are simply talking about agent modeling and everyone knows that they are so limited.

SALV.: You may know mathematics well, but you don't know science by just looking at real world. Non-linear equations for macro-variables, exact solutions, stability and chaos may be fun to play with. But they are made-up plays only.

SAGR.: I agree with Salviati. Think of molecules. They interact with each other, optimizing action, which may be comparable to the notion of utility function in economics. With complete understanding of such interactions at micro-level and introducing concepts of statistical physics on how to handle the large number of such interacting heterogeneous objects comes understanding of the dynamics of macro-matter.

SALV.: Such was elaborated upon in Chapter 1.

SALVIATI: Greetings, Sagredo, and Simplicio, my good friends. Yesterday, we resolved to meet today and discuss as clearly and in as much detail as possible the character and the efficacy of those laws of macro economics, which up to the present, have been put forth by the books of Aoki and Yoshikawa (2006) and Aoyama et al. (2010a), the very same authors of this book.

SAGREDO: Indeed, I am truly glad and honored to meet you and Simplicio on this occasion of the completion of this book.

SIMPLICIO: Indeed, indeed, (with a touch of doubt on his face) what is “macro-econophysics”?

SALV.: You must be familiar with “econophysics”. It is evidence-based economics as a science. You may recall that it has the word “physics” in it as many physicists have devoted their research to this area, guided by the concepts and ideals of physics in their heart.

SAGR.: I see that most of the authors are physicists, except for Prof. Yoshikawa, who I heard is a macro-economist.

SALV.: They both have put forth the same ideals of revolutionizing the way real economy is studied in their respective books before. Now they have joined forces to introduce the term Macro-econophysics.

SAGR.: I have heard that it shares its ideals with “agent-based modelling”, which is yet another great approach.

SIMP.: That is good. But isn't this book a mere collection of the respective topics from each of the authors?

SALV.: Absolutely not. They have been working together for the last few years, combining the best of physics and economics and publishing papers. They have spent many days and nights discussing all things big and small included in this book.

SAGR.: And look… they have Professor Richard Feynman's very hopeful words on the front cover.

SALV.: This book is one of the latest efforts to construct a science of economics, which forms a part of the basis of this new development and will be improved and passed on to the next generation of academics.

SAGR.: And look at this photograph taken by the first author (who, by the way, is the first author because of the names being listed in alphabetical order) on December 1, 1979, at UC Irvine, California. He and his famous van!

What causes business cycles? A look at Haberler (1964) shows that all kinds of theories had already been advanced by the end of the 1950s. In recent years, macroeconomists had been so confident of their skillful policy management as to hail it as the “Great Moderation”. Lucas Jr. (2003), in his presidential address to the meeting of the American Economic Association, declared:

Ironically, when Lucas delivered this address, the American economy had been on a steady road to the worst post-war recession, perhaps even depression. Business cycles are still with us and await further investigation.

Describing the characteristics of business cycles has been one of the most difficult problems in the study of economies because of the presence of various economic shocks. In this chapter, we analyze the characteristics of business cycles more directly than in the standard approach using a method motivated by physics. First, we identify the causes of business cycles using the random matrix theory and principal component analysis.

We then identify evidence of synchronization in Japanese business cycles and the international business cycle. Finally, we discuss the mechanism of this synchronization using the coupled limit-cycle oscillator model.

What Causes Business Cycles

Business cycles are defined as a type of uctuation in the aggregate economic activity of nations that organize their work mainly around business enterprises. A cycle consists of expansions occurring at roughly the same time in many economic activities, followed by similar general recessions, contractions, and revivals that merge into the expansion phase of the next cycle; it is a sequence of change that is recurrent but not periodic.

Background and Motivation

Macroeconomics is aimed at understanding the behavior of the economy as a whole— business cycles, economic growth, employment/unemployment, in ation/de ation, and inequality of income. A macroeconomy, such as that of the U.S. and Japan, consists of more than 100 million consumers and one million firms. Evidently, the behavior of the macroeconomy is the outcome of the aggregation and interactions of a large number of micro units such as consumers and firms. To understand such interactions beneath the surface of the macroeconomy, one must resort to new methods that are different from those of standard microeconomics. We propose that this new method be called Macro-econophysics. It provides one not only with novel theoretical insights but also with new empirical methods. Following the spirit of physics and other natural sciences, macro-econophysics puts equal importance on theory and on empirical findings. This book is concerned with both macroeconomic theory and important findings that advanced the study of macroeconomics.

Macroeconomics was born in the first half of the 20th century. Above all, it was expected to make a good diagnosis of, and hopefully provide a prescription for depression and high unemployment. Keynes’ General Theory of Employment, Interest, and Money, published in 1936, was a landmark work, and had such a profound impact on the discipline that after the war, macroeconomics became synonymous with Keynesian economics (Keynes, 1936). Paul Samuelson once proposed “the neoclassical synthesis,” wherein it is held that the achievement of full employment requires Keynesian intervention but neoclassical theory is valid and useful for efficient resource allocation when full employment is achieved. To many, it was then obvious that the economy occasionally lapses into a recession where labor and other resources are left unemployed. The usefulness of Keynesian economics was taken for granted. Thus, during the 1950s and 1960s, the profession accepted the neoclassical synthesis, and held that economics stood on two pillars, namely, neoclassical microeconomics and Keynesian macroeconomics.

However, Arrow (1967), in his review of the Collected Scientific Papers of Paul Samuelson, pointed out that the neoclassical synthesis was actually nothing but a common sense argument and lacked rigor.

This chapter explains basic concepts and analytical methods extensively used in subsequent chapters. Readers begin with learning fundamental stochastic modeling to reproduce distributions with power-law tails. Such skewed distributions are discussed extensively in Chapter 3, with important examples of income and firm-size distributions. Next, we explain entropy. Readers must have already recognized that the concept of statistical equilibrium and probabilistic description, developed in physics, would be indispensable for understanding complex macroeconomic phenomena. Mechanical equilibrium cannot adequately accommodate the diversity of an economic system. Entropy, which measures the degree of randomness or disorder in a system, is a major player in thermal and statistical physics. The statistical equilibrium of a system is regarded as a manifestation of the maximum entropy principle. We first reiterate the basics of entropy especially for readers who have no physics background; this serves as a prelude to Chapter 4 that deals with the distribution of labor productivity. Finally, we provide a brief account of the stochastic modelling of stationary and non-stationary time series with a avor of nonlinear physics. This part is expanded in Chapters 5, 6, 7, and 8 on multivariate time series analysis, business cycles, collective motion of prices, and correlation and synchronizing networks, respectively.

Stochastic Models and Fat Tails

Many phenomena in socio and economic systems exhibit fat-tailed and skewed distributions. We have observed and will observe this fact for personal income, firm-size, productivity dispersion, economic networks, and so forth in this book. It is often called “A few giants and many dwarfs”. Presence of a few giants and comovement of many dwarfs has important consequences to a macroeconomic system as we shall explain at places in this book.

Natural science has also witnessed many phenomena, notably in the study of fractals and scaling, and attracted many researchers who found models and scenarios specific to each domain, and even speculated “universal” explanations for the origin of such distributions, especially of power-laws that possess mathematical properties of scale-free.

In fact, one of the oldest and most widely known models is the so-called Yule's model to explain distribution of biological species.

As pointed out in the Introduction, a macro economy consists of many micro agents. These agents are not independent of each other, but form networks, i.e., they have relationships among each other. An understanding of their interactions is crucial for an understanding of a macro economy.

This chapter explains the basic concepts and techniques for understanding the structure of economic networks, and then applies them to economic networks. Economic networks are not random nor regular, but exhibit fat-tailed distributions for the numbers and weights of relationships. They also form densely knit groups that are only loosely connected with each other, called clusters or communities.

Basic tools cover