Investigations over more than a century and the recent availability of electronic databases of income and wealth distribution (ranging from a national sample survey of household assets to the income tax return data available from governmental agencies) have revealed some remarkable features. Irrespective of many differences in culture, history, social structure, indicators of relative prosperity (such as gross domestic product or infant mortality) and, to some extent, the economic policies followed in different countries, the income distribution seems to follow a particular universal pattern, as does the wealth distribution: after an initial rise, the number density of people rapidly decays with their income, the bulk described by a Gibbs or log-normal distribution crossing over at the very high income range (for 5–10% of the richest members of the population) to a power law, as shown in Fig. 1.1. The power law in income and wealth distribution is called the Pareto law, after the Italian sociologist and economist Vilfredo Pareto. The log-normal part is named as the Gibrat law, after the French economist Robert Gibrat. This seems to be a universal feature: from ancient Egyptian society (Abul-Magd 2002) through nineteenth-century Europe (Pareto 1897; Champernowne 1953) to modern Japan (Chatterjee et al. 2005b; Chakrabarti et al. 2006). The same is true across the globe today: from the advanced capitalist economy of USA (Chatterjee et al. 2005b; Kar Gupta 2006a; Richmond et al. 2006) to the developing economy of India (Sinha 2006).

In the earlier chapters, we introduced kinetic exchange models and discussed the mathematics behind them. However, one major point that is missing in the earlier discussions is the choice behaviour of the agents. The outcomes of the stochastic process do not reflect any optimization mechanism on the part of the agents. In this chapter, we will provide a simple economic model which intends to capture the basic features of the kinetic exchange models. We start with some usual assumptions regarding the preference pattern of the agents and the market mechanism. Eventually, it will be shown that the outcomes are exactly the same as those obtained in the kinetic exchange models, thus providing an elementary (but non-unique) way of interpreting the stochastic money evolution equations in economic terms. After that, we will discuss the dynamic features of the asset distribution in the economy if it has time-varying macroeconomic variables. To be precise, we will discuss a possibility of inequality reversal (as has been postulated and discussed in Kuznets (1955, 1965) and Angle (2010)) in the same framework.

Derivation of the basic kinetic exchange model

Following Chakrabarti and Chakrabarti (2009), we consider an N-agent exchange economy in discrete time. At every point of time, exactly two agents are randomly chosen from the pool of N agents, i.e. each agent has equal probability of being chosen for trade. The exact trading mechanism is described below. After they trade, the agents part and leave the market. In the next period again two agents are chosen for trade and the same process is repeated until the distribution of their assets reaches a steady state.

In some of the earlier chapters, we have presented in detail the models which have been inspired from physical theories. In the last chapter, we presented a simple economic model which intends to capture the basic features of the kinetic exchange models. The objective of this chapter is to present analytical discussions on stochastic and related economic models of the distribution of income and wealth.

The economic significance

In order to get an idea about the distributional effects of a particular economic policy it often becomes necessary to have information on the distribution of income. Also inequality based on distribution of income has important effects on development, social outcomes and public finance. The shape of the income distribution in a country enables the policy-makers to get an idea about the amount of tax collection. For instance, in Germany 50.6% (19.7%) of the entire income tax is paid by the top 10% (1%) of the taxpayers (see Merz et al. 2005).

There are economic and non-economic reasons for separate study of the distribution of wealth that cannot be interpreted as human capital, such as educational background. Examples of wealth of this type are financial assets and real properties. Unlike human capital they can be traded in appropriate markets at the time of necessity; for instance, when the flow of regular income reduces (after retirement) and when consumption is likely to increase with an increase in family size. Precautionary motive for saving is also regarded as a major reason for accumulation of wealth. Wealth accumulation is often taken as an indicator of social status. Wealth may be retained as well for the purpose of bequests.

It would be difficult to find any society or country where income or wealth is equally distributed among its people. Socioeconomic inequality is not limited to modern times; it has been a persistent fact, and a constant source of irritation to most, since time immemorial.

The issue of inequality in terms of income and wealth is perhaps the most fiercely debated subject in economics. Economists and philosophers have spent much time on the normative aspects of this problem (Rawls 1971; Scruton 1985; Sen 1999; Foucault 2003). The direct and indirect effects of inequality on society have also been studied extensively. In particular, the effects of inequality on the growth of the economy (Benabou 1994; Aghion et al. 1999; Barro 1999; Forbes 2000) and on the econopolitical scenario (Blau and Blau 1982; Alesina and Rodrik 1992; Alesina and Perotti 1996; Benabou 2000) have attracted major attention. Relatively less emphasis has been put on the sources of the problem itself. There are several nontrivial issues and open questions related to this observation: How are income and wealth distributed? What are the forms of the distributions? Are they universal, or do they depend upon specific conditions of a country? Perhaps the most important question is: if inequality is universal (as some of its gross features indicate), then what is the reason for such universality?

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.