Economics has emerged as a major discipline today of interest to all because of its impact on our day-to-day life. What has been achieved so far has been truly impressive, although the discipline is not as successful as one would expect. Notwithstanding what mainstream economics does or strives for, it does not really meet the criteria to be called a natural science yet. This book is an attempt to steer it in that direction.

Although natural sciences, such as physics, chemistry, biology, or geology employ logic and mathematics (as a condensed form of logic), it is never the sole ingredient. Stepwise observations are organized in a logical fashion, often with the help of tentative or approximate hypotheses, and both the existing observations and predicted outcomes are carefully compared. The understanding of the next level or of similar but different systems grows progressively, based on the successful ideas or an understanding developed earlier. Naturally, there is interdependence in natural sciences as a consequence of this kind of development. In general, precise knowledge, successful ideas, or techniques developed in one area of the natural sciences become easily translated into another.

This interdependent structure of research in the natural sciences also gets reflected in the graduate level course structure for students in their respective disciplines. Students of one major discipline of the natural sciences have to learn the basic and established concepts in other disciplines: Physics majors learn concepts of chemistry, biology, or geology; biology majors learn basic concepts of physics and chemistry, along with others. This practice is somehow not there yet for the social sciences; the graduate students here do learn mathematics and statistics but not the basic concepts of physics, chemistry, or biology. Personal interests are, of course, exceptions and are not counted here!

To many, this is the main reason why economics, which also started becoming formalized much later compared with most other branches of the natural sciences, could not boast of the spectacular successes achieved by other disciplines. Among others, econophysicists believe in the need for a similar mutation of ideas from economics and physics, for the healthy evolutionary growth in both.

This chapter provides the rationale of the book together with a reading road-map. Section 1.1 identifies the historical and methodological reasons for the current impasse in macroeconomics, while Section 1.2 offers some remarks about the issue of aggregation. The motivation for modelling economies with heterogeneous interacting agents, the necessity of coupling ACE (agent-based computational economics) and ASHIA (analytical solution to heterogeneous interacting agent-based models) and the ME (master equation) approach that the book suggests are discussed in Section 1.3. Finally, Section 1.4 details the structure of the book.

Why are We Here?

Magma, something in between solid and liquid states, describes well the state of macroeconomics today. Since after the Great Recession, it has been possible to find reports of deep states of disaffection (Solow, 2008) vis a vis comfortable views (Blanchard, 2008). However, the more relaxing approach has caveats of such a magnitude to alert even the quietest reader. According to their proponents, two (out of three) equations of the new Keynesian dynamic stochastic general equilibrium (DSGE) are manifestly wrong, while the methodology based on the representative agent (RA) cannot be reconciled with the empirical evidence. Lakatos (1978) would have described the present state of mainstream macroeconomics as having clear signs of a failing paradigm.

Economics is slowly moving from the classical physics to the statistical physics paradigm, from determinism to stochasticity, from equilibrium to interaction and complexity. That is a passage, from the isolated not interacting individual to a stochastically interactive framework, which generates an emergent macroeconomics (Delli Gatti et al., 2008).

Although speaking the language of probability and stochastic processes theory, which is familiar to economists, this book argues for the adoption of tools widely developed in the field of statistical physics. The introduction of this approach is not without consequences in the corpus of economic thought. As will be clearer in what follows, this modifies the characteristics of the equilibrium and the interpretation of dynamics, implying a change in the economic paradigm. The Great Recession is not due to mainstream economics virtues attributed to the market theory; but it has been worsened by a bizarre theoretical interpretation of the markets.

More than ten years ago, Frank Hahn and Robert Solow, two of the most influential economists of the twentieth century, delivered a fierce attack against the main trend of macroeconomic theory and the methodological quest for micro-foundations by arguing against […] the belief that the only appropriate micro model is Walrasian or inter-temporal-Walrasian or, as the saying goes, based exclusively on inter-temporal utility maximization subject only to budget and technological constraints. This commitment, never really argued, is the rub (Hahn and Solow (1995), p.1).

Admittedly, their criticisms and proposed solutions have gone largely unnoticed. The alternative proposal we shall put forward in what follows with the Walrasian paradigm is definitely more radical. As a preliminary step, however, it seems worthwhile to emphasize that Walrasian micro- foundations should be considered as a wrong answer to what is probably the most stimulating research question ever raised in economics, that is, to explain how a completely decentralized economy composed of millions of (mainly) self-interested people manages to coordinate.

The way in which billions of people, located everywhere in the world, exchange goods, often coming from very remote places, in decentralized market economies is a fundamental problem of macroeconomic theory. At present, there is no consensus regarding which theory best explains this problem. At one end of the spectrum, new classical macroeconomists work within frameworks in which the macroeconomy is assumed to be in a continual state of market clearing equilibrium characterized by strong efficiency properties. At the other end of the spectrum, heterodox macroeconomists argue that economies regularly exhibit coordination failure (various degrees of inefficiency) and even lengthy periods of disequilibrium. Given these fundamental differences, it is not surprising to see major disagreements among macroeconomists concerning the extent to which government policy makers can and ought to attempt to influence macroeconomic outcomes.

This chapter introduces master equation (ME) modelling, which is widely used in hard sciences and encompasses pure probability, statistical physics and chemistry. The development of the ME involves a great deal of mathematical tools. However, this chapter introduces a very restricted set of basic notions and presents an even more restricted set of tools, in order to introduce the basic principles and provide a sufficient intuition. It highlights the potential of this approach to complex systems analysis and, in particular, for dealing with heterogeneity and interaction of agents.

Section 3.1 provides an introduction by recalling the notions introduced in Chapter 2. Section 3.2 introduces the ME from a very generic point of view: it shows how the ME can analytically describe processes in a discrete-space and continuous-time framework. Section 3.3 provides further developments in the case of Markov processes. In this section, the ME is introduced from the easiest case to the most general one, providing a short description of the stationary solution. Section 3.4 is devoted to the solution of the inferential problem of identifying the moments of the stochastic process described by the ME. The results presented in this section are of particular relevance with reference to the applications developed in the later chapters of Part II. This section provides the reader with the necessary background for understanding the analytical results developed in the application chapters, which involve more sophisticated techniques developed in Appendix B and C. The last Section 3.5 concludes.

Introduction

Chapter 2 introduced the notion of occupation numbers in equation (2.13) to evaluate how many units belong to a given class or subsystem of reference. Accordingly, the components of the occupation vector N(t), defined by equation (2.15), describe how agents of the system distribute over a set of states, classes or among subsystems. This distribution is given by the occupation numbers Nh(t) for each group indexed by h and defines the configuration of the system.

Introduction

The first notion to consider is the notion of system. Intuitively, a system can be defined as an assembly of parts, as a collective body, where each element does something in such a way that, all together, the parts cooperate to perform some kind of collective action or cause a macroscopic emergent phenomenon. From a complex systems perspective, a system is a collection of constituents dominated by intrinsic stochastic laws, due to the interactive behavior of heterogeneous parts, for every possible granularity of the description.

If the composition into parts is relevant, the system is the ensemble of subsystems, a macroscopic collective body of microscopic units which organize a hierarchical structure of mesoscopic subsystems. The granular composition of a macro-system is mainly relevant when the ensemble of micro-systems is dominated by heterogeneity and interaction inducing emergent phenomena which cannot be explained other than by following statistical reasonings and inferential methods, that is, phenomena taking place at the macroscopic level and for which the complete knowledge of the microscopic world is not enough.

For instance, assume that the system is a sector of economic activity in the economy and consider there are firms of different sizes, for example, small, medium and big firms. Three ensembles can be isolated, which are defined as subsystems. Although each of these groups is a collective body, none is as macro as the economy (the complex system) and none is as micro as the single firms (the elementary constituents). This is the mesoscopic perspective: between the very macro-system and the very micro-systems, there can be a wide spectrum of meso-systems.

A system is therefore not an absolute notion; it involves a hierarchy of organized structures made up of constituents at lower levels of aggregation.

At the micro-, meso- and macro-level, observables can be conceived as (random) variables in space and time. Observables can be represented as a set of quantities, a family of measurable quantities on a system made of N observation units.