Anwar Shaikh Some Universal Patterns in Income Distribution: An Econophysics Approach August 2018 Working Paper 08/2018 Department of Economics The New School for Social Research The views expressed herein are those of the author(s) and do not necessarily reflect the views of the New School for Social Research. © 2018 by Anwar Shaikh. All rights reserved. Short sections of text may be quoted without explicit permission provided that full credit is given to the source. Some Universal Patterns in Income Distribution: An Econophysics Approach Abstract This paper utilizes the econophysics "two class" (EPTC) approach to income distribution to derive certain empirical rules applying to all countries in the comprehensive World Income Inequality (WIID) database. This approach demonstrates that wage incomes follow an exponential distribution while property incomes follow a Pareto distribution, which leads to a simple and empirically robust approximation to the Lorenz curve. We in turn show that the per capita income of any bottom fraction (x) of the population is proportional to “inequality adjusted GDP per capita”, i.e. to (GDP per capita)∙(1-Gini), the constant of proportionality a(x) being solely a function of population fraction under consideration. This proposition is empirically robust across countries and over time in our large database. We focus on two patterns. The “1.1 Rule” in which the income per capita of the bottom 80 percent of a country's population, what we call the Vast Majority Income, can be calculated in every country as 1.1(GDP per capita)∙(1-Gini). Using the VMI in place of GDP per capita gives rise to different country rankings. Secondly, the “1.0 Rule” in which the per capita income of the bottom 70 percent is directly equal to inequality adjusted GDP per capita. Sen (1976) uses traditional welfare theory to arrive at inequality adjusted GDP per capita as a measure of social welfare while we use EPTC to arrive at it as a measure of the per capita income of the bottom 70 percent. Keywords income distribution, inequality, econophysics, economic indicators, world, international JEL Classification : D31, D63, N30, O50 Some Universal Patterns in Income Distribution: An Econophysics Approach Introduction The analytical treatment of income distribution goes back to Pareto’s (1897) demonstration that the distribution of incomes of top earners can be characterized by a power law now known as the Pareto distribution. Over the last decades the path-breaking works of Piketty, Atkinson and Suez (2013) have sparked a great resurgence of interest in income inequality on national (Cingano 2014, Ravallion 2014) and global levels (Alvaredo et al. 2018, Milanovic 2016), and in the difference between the distributions of wage and property incomes. More recently, in the spirit of Pareto’s original inquiry, the econophysics "two-class" (EPTC) theory of income distribution pioneered by Yakovenko and his co-authors (Dragulescu and Yakovenko 2000, 2001, Yakovenko and Rosser 2009, Banerjee and Yakovenko 2010) has opened new ground (Rosser 2006; Yakovenko and Rosser 2009). Their theoretical foundation is a kinetic approach in which income from wages and salaries is characterized by additive diffusion1 which leads to an exponential distribution, while income from investments and capital gains is characterized by multiplicative diffusion which leads to a Pareto distribution (Silva and Yakovenko, 2004, p. 6). The overall Lorenz curve is then comprised of a large section corresponding to the exponential distribution and a small one (roughly the top 3 percent of the population) corresponding to the Pareto, which leads to a simple and robust approximation to the whole Lorenz curve that fits income distributions in many advanced countries (Nirei and Souma 2007; Derzsy, Néda, and Santos 2012; Jagielski and Kutner 2013; Shaikh, Papanikolaou, and Wiener 2014; Shaikh 2016; Oancea, Andrei, and Pirjol 2016, Tao et al 2017). 2 Some Universal Patterns in Income Distribution: An Econophysics Approach Concerns about income distribution often focus on the average (i.e. per capita) incomes of a subsets of the population. Poverty measures typically concentrate on the incomes of bottom cohorts such as the poverty rate measures used by international development institutions (UN 2016) or the shared growth concept adopted by the World Bank which focuses on the income growth of the bottom 40 percent (World Bank 2016). On the other end, considerable attention has been given to the incomes of top 20, 10 or even top 1 per cent of the population (Atkinson and Pickett 2013). In this paper we utilize the EPTC framework to demonstrate that per capita incomes of the various subgroups of the population in a country are proportional to the product of two variables alone: GDP per capita (y) and (1 – G) where G is the Gini coefficient, with a constant of proportionality which depends solely on the subgroup under consideration. For instance, in the case of per capita income the bottom 80 per cent (y (80)), which we call the Vast Majority Income (VMI), the theory predicts a constant of proportionality in the range 1.15 – 1.20, for 1-Earner and 2-Earner households respectively, while for the per capita income of the bottom 70 percent (y(70) the predicted coefficient would be in the range 0.97-1.01, i.e. essentially 1 (Table 1). The latter implies that the per capita income of the bottom 70 per cent is simply yG(1− ). It is interesting to note that Sen (1976) used traditional welfare theory to derive a measure of social welfare which has been interpreted as an inequality-adjusted measure of per capita income (UNDP 1991, 1994). From our point of view, using Sen's measure is equivalent to assessing national progress in terms of the per capita income of the bottom 70 percent. Yet we arrive at this measure and its properties from an entirely different 3 Some Universal Patterns in Income Distribution: An Econophysics Approach yy− (70) foundation. The previously noted relation y(70) =− y( 1 G) also implies thatG = , y i.e. that the Gini coefficient is a measure of the distance between average per capita income and that of the bottom 70 percent. Finally, we show in the theoretical section of the paper that the Gini coefficient is a linear function of the share of property income in net national income, which turns out to have important implications for the interpretation of Piketty’s argument (Shaikh 2016). At an empirical level, we show from data available in the World Income Inequality Database that our theoretical results are empirically robust across all available countries over all available years. For example, the per capita income of the bottom 80 per cent of the population (VMI) has an empirical constant of proportionality 1.1 which is close to the theoretically predicted range, while the per capita income of the bottom 70 per cent has a coefficient of 1.0 which is the essentially the theoretical one. These findings are remarkably accurate in every country from Norway to Niger and in all years from 1977-2014. Finally, while the EPTC theory does not privilege any particular per capita income, we show that the VMI has special theoretical and empirical properties which enhance its obvious social and political significance. Econophysics and income inequality Econophysics is one of the older branches of analytical economics (Rosser and Yakovenko 2009). It is widely acknowledged that "income and wealth distributions of various types can be obtained as steady-state solutions of stochastic processes" (Kleiber and Kotz, 2003, p. 4 Some Universal Patterns in Income Distribution: An Econophysics Approach 14). But there is little agreement as to which probability distribution functions (pdf) best characterize the available data. A recent approach within "econophysics" has been to characterize the overall income distribution as the union of two distinct pdfs, with the exponential curve applicable to first 97-99 percent of the population of individual-earners and the Pareto or some other power law applicable to the top 1-3 percent (Dragulescu and Yakovenko, 2002, pp. 1-2). The theoretical foundation for this "two-class structure of income distribution" is a kinetic approach in which income from wages and salaries yields additive diffusion while income from investments and capital gains yields multiplicative diffusion (Silva and Yakovenko, 2004, p. 6). This leads to an approximation of the overall Lorenz curve as a weighted average of an exponential curve applicable to the vast bulk of the population, and a fixed term which kicks-in at the highest level in order to account for the Pareto tail (Silva and Yakovenko, 2004, Abstract). The approximation takes advantage of the fact that the population percentage at the higher end is very small but its income fraction (f) is nonetheless significant. This formulation gives rise to a strong relation between income and inequality which proves to be empirically very robust against a large sample of international data. The first step in this direction is to note that the per capita income y (x) of any proportion of the population reflects both the average per capita income and the degree of inequality. th If we designate the population and income of the i fractile (quintile or decile) by Xi and Yi n X = respectively, then for the economy as a whole total population X i , total income i=1 5 Some Universal Patterns in Income Distribution: An Econophysics Approach n Y = and per capita income YX. Let x be some cumulative percentage of the Y i y = i=1 x population to which corresponds a total population X( x) = X i , total income i=1 x ( ) = Y x X x Y() x= Yi and per capita income y x ( ) ( ) . But the cumulative population i=1 x X proportion is itself x = i and the corresponding cumulative income proportion is i=1 X x Y yx( ) = i .