Scaling distributions in economics: The size distribution of incomes

In economic data, skewed and thick tailed frequency curves are the rule rather than the exception. Again, there seems to be an increasing interest in the study of economic variables which can be considered as realizations of a stochastic process. Within this class a major role is played by income distribution. It is undoubtedly true that the distribution of the command over the goods and services produced by a society is of crucial importance for the student of welfare economics. On a more practical level, producers of consumer goods must study the distribution of income in order to estimate the probable extent of their markets. The size distribution of income also affects the social and economic policy-making when planning the magnitude and the structure of taxation schemes or evaluating the effectiveness of tax reforms. In economic and social statistics, the form of the income distribution is the basis for the measurement of the inequality of incomes and more general social welfare evaluations. Over the last 100 years, a large number of distributions has been proposed for the modeling of size distribution of personal incomes. The most canonical version of such models was discovered by the mathematical economist and sociologist Vilfredo Pareto in 1897. Pareto formulated his model to explain the distribution of income and wealth, and believed that it was a universal candidate for the mathematical description of given income data for different countries over relatively long time periods. However, more recent studies suggest that it is only the upper end of income and wealth distributions that follows the Pareto's model, with the lower ends following the lognormal form of the Gaussian distribution that is associated with the random walk, originally proposed for the whole of the income distribution by the French economist Robert Gibrat in 1931. Since the early 1990s, there has been an explosion of work on economic size phenomena in the physics literature, leading to an emerging new field called "econophysics". The list of such phenomena includes, among others, the distribution of returns in financial markets, the distribution of income and wealth, the distribution of economic shocks and growth rate variations, and the distribution of firm sizes and growth rates. A common theme among those who identify themselves as econophysicists is that standard economic theory has been inadequate or insufficient to explain the non-Gaussian properties empirically observed for various of these phenomena, such as excessive skewness and leptokurtotic fat tails. Indeed, many economic phenomena occur according to distributions that obey scaling laws rather than Gaussian normality. Whether symmetric or skewed, the tails are fatter or longer than they would be if Gaussian, and they appear to be linear in figures with the logarithm of a variable plotted against the logarithm of its cumulative probability distribution. This is true of the Pareto's distribution, and soon a variety of efforts have been made by physicists, mathematicians, and economists to model a variety of stochastic economic phenomena using either the Pareto's distribution or one of its relatives or generalizations, such as for instance the Lévy stable distribution. In the light of these considerations, this chapter offers a reader's guide to "scaling" phenomena in economics with particular focus on the interplay between the dynamics and empirical regularities of income distribution.