Why does this imbalance exist in the first place? Why are a few rich and many poor? For centuries we have borne the effects of this inequality. We know neither the cause nor the solution to this elusive problem. From philosophers to economists, many have vehemently tried for ages to understand the reasons and formulate remedies for such inequalities. No doubt, great efforts have been made to tackle this multifaceted problem, but the situation has been analogous to fighting the Greek mythological monster Hydra, who grows two heads in place of an injured one. Overcoming this problem, indeed, seems to be a Herculean task!

Heraclitus said, ‘change is the only constant’. Putting our faith in him, one might have expected things to change drastically, and the inequality to even disappear at some point in time! Strangely, this has not been the case. We find that inequality has been a universal and robust phenomenon – not bound by either time or geography. Fortunately for scholars, it has a few statistical regularities, most of which have been recorded in the past 115 years or so. Owing to the seminal works of Pareto (1897) and Gibrat (1931), one can now identify certain regularities in the income and wealth distributions over a wide range of societies and time periods. Physicists have come up with some very elegant and intriguing kinetic exchange models in recent times to shed some light on these observations. Our intention is to describe these developments in this book.

The kinetic exchange models of markets provide a simplified – perhaps oversimplified – picture of the exchange mechanism that takes place in a real market. However, the simple, “toy model” paradigm also offers interesting grounds for possible analytic formulation, compared with real markets where the dynamics involve plenty of parameters and involve complex evolution that render them intractable and incomprehensible.

In this chapter we will discuss in detail some of the simple and intuitive frameworks developed to understand the qualitative, and sometimes even the quantitative, aspects of simple kinetic exchange models discussed in the previous chapters. Most of our discussions will include the CC (Chakraborti and Chakrabarti 2000) and the CCM (Chatterjee et al. 2004) models, and some of their important variants, which are easy to handle analytically, or in cases where their solutions can be argued intuitively.

Analytic results for the CC model

The earliest attempt to understand the CC model analytically (Das and Yarla–gadda 2003) assumes that, independent of the initial conditions, the system evolves to an equilibrium distribution after a suficiently long time. Thus, in the steady state, the joint probability that, before interaction, money of i lies between x and x + dx and that of j lies between z and z + dz is f(x)dxf(z)dz.Since each interaction conserves total money, let L = x + z.

Throughout the recorded history of human civilization, we have witnessed the bitter outcomes of economic inequality – social tensions, conflicts, etc. This incessant problem has been addressed by some of the greatest thinkers, philosophers and social scientists, including economists. Questions on the nature of the distributions of wealth and income have been raised repeatedly. More so, during or just after periods of crisis, wars and social calamities. In this book, we have tried to present a new interdisciplinary approach in analysing and dealing with the age-old problem of economic inequality in the societies. This paradigmatic shift has been possible owing to the combined efforts of economists, mathematicians and physicists (Cockshott et al. 2009; Sinha et al. 2010).

Noting that this inequality has a very robust and universal statistical form (discussed extensively in the first two chapters of this book), and the fact that some core human ability factors, such as the intelligence quotient or health factors, are distributed according to the normal (or Poisson, at times) distribution, a natural question to ask is why are the distributions in wealth and income so different from the normal? Why do they have such broad distributions, and with ubiquitous power law tails? The very fact that these distributions have such different characteristics (and are universally observed) indicates that there must be a deeper cause (and a common underlying mechanism).

In this chapter, we discuss the background and motivation for various modelling efforts carried out over the years. While some models take into account the various details as in real economic transactions, trying to fit the detailed characteristics of empirical data, others reproduce the empirical features qualitatively using very simple ideas of physical origin.

Models of income distribution

Pareto's extensive studies in Europe showed that the tail of the wealth distribution for the richer sections of society follows a power law (Pareto 1897), now known as the Pareto law. Gibrat (1931) separately worked on the same problem and proposed a law of proportionate development. Much later, Champernowne also considered this problem systematically and proposed a probabilistic theory to justify Pareto's claim (Champernowne 1953; Champernowne and Cowell 1998).

Gibrat (1931) observed that the power law distribution did not fit all the income data and hence proposed a law of proportionate effect, which states that a small change in a quantity is independent of the quantity itself. Thus the distribution of a quantity dz = dx/x should be Gaussian, and hence x is log-normal. The asymptotics of this can lead to the Pareto power law. Gibrat found that the small-and middle-income ranges appear to be well described by a log-normal distribution (see Eq. (2.3)). The details of Gibrat's law and its derivation will be presented in Chapter 7.