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.