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).
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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 yG(1− )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
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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.
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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
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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 . It follows that the per capita income of the bottom x percentage relative to i=1 Y the national average IR ( x)is equal to the ratio of the cumulative income proportion of the bottom x percentage of the population to x itself. This means that we can calculate the per capita income of (say) bottom 80 percent of the population simply by summing relative incomes up to 80 percent and dividing this by 0.80.
xx YX xx ii YXiiii==11 (1) IR(x) y(x) y = = = y( x) x ii==11YXYX( )
Since the y-axis of the Lorenz curve is the cumulative income proportion yx( )and the x- axis is the cumulative population proportion x , the relative per capita income ratio IR(x) of bottom x percentage of the population is simply the slope of the ray through the origin to the point on the Lorenz curve which represents the population proportion x. In the case of the 80 decile, the slope of the line C in Figure 1 represents the IR for the bottom 80 percent IR(80). It follows that IR ( x)is a measure of inequality since it depends crucially on
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Some Universal Patterns in Income Distribution: An Econophysics Approach
the shape of the Lorenz curve. Indeed, it is similar to (1-G), which is the ratio of area beneath the Lorenz curve in Figure 1 to the area beneath the 45-degree line A2, since both equal 1 under perfect equality, and both equal 0 under perfect inequality. While (1-G) captures the whole shape of the Lorenz curve, IR(x) only samples it at a single point. There is therefore no a priori reason to expect a stable relation between the two. Yet this is precisely what econophysics claims.
[Figure 1]
To elucidate the EPTC argument we need to consider the Lorenz curve as a combination of two distinct distributions. Let yx' ( ) = the cumulative (pre-tax) income share stemming from the exponential section of the overall Lorenz curve, G ' = the Gini Coefficient of the exponential portion of the overall curve, f = the proportion of total income in the Pareto section (the share of total property income), and ( x −1) = the step function such that
= 0for x < 1 (i.e. along the exponential section) and = 1for x = 1 (along the Pareto section, which is approximated by a vertical line at x = 1). Then the overall Lorenz curve and the corresponding overall Gini coefficient3 can be expressed as
(2) y( x) = y'1( x)(1 − f ) + ( x− )
(3) (1−G) =( 1 − G ')( 1 − f )
Along the exponential section of the Lorenz curve the step function (x −1) = 0, so equation (2) reduces to yy( x) =−'( x)(1 f ), which combined with equation (3) can be
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Some Universal Patterns in Income Distribution: An Econophysics Approach
y()/ x x y' ( x) x written as = . But we saw earlier in equation (1) that the left-hand side (1−−GG ) 1 ' of this relation is simply IR(x) y(x) y , the ratio of the relative per capita income of the bottom x percentage of the population to that of the whole population. Hence
yy(x) y'( x) x (4) = ax(), where ax() (1− G ) 1'− G
This is a very powerful result because for an exponential pdf the Gini coefficient is a constant and the cumulative income proportion yx' ( ) is a parameter-free function of the cumulative population proportion x (Silva and Yakovenko, 2004, pp. 1-5), so that the term
in equation (4) is solely a function of x. The per capita ratio
IR(x) y(x) y was previously shown reflect the degree of inequality inherent in a Lorenz curve, and to have the same limits as the equality index (1 − G) . But now we see that the former is actually proportional to the latter through a constant of proportionality
y' ( x) x ax( ) = which depends only on x. It follows that within the exponential section 1'− G
(which is roughly 97 percent of the population) the per capita income of any cumulative population percentage x is proportional to the inequality-discounted average income per capita. As shown in Equation (5), the percentage change in any subgroup per capita income is a sum of percentage changes in average income per capita and (1-G).
y' ( x) x (5) yy(x) = ax( ) (1 − G) , where ax( ) = 1'− G
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Some Universal Patterns in Income Distribution: An Econophysics Approach
The EPTC approach considers two types of populations for the lower part of the income distribution: individual earners and two-earner households. In the former case, the Gini
' coefficient is G1 = 12 and the Lorenz curve is y= x +(1 − x) ln( 1 − x) ; in the latter case, which they argue is a good representation of overall family income, the Gini coefficient is
' G2 = 35 and the Lorenz curve is an implicit function y( x) = f( x) since both y and x are functions of household income (Dragulescu and Yakovenko, 2001, pp. 585-588). The
y' ( x) x proportionality parameter ax( ) = can be directly calculated in either case, as 1'− G shown in Figures 2A-2B and Table 1 which lists the values for x from 0.70 to 0.90 since this is the region of our concern. Also displayed here are the actual empirical ratios for these three deciles as calculated from the large international sample discussed in the next section.
The empirical ratios for all deciles can be found in Figure 3. Since the empirical evidence is in terms of "personal-equivalent" units derived from household data (see the Data
Appendix), the relevant theoretical distribution is that of household incomes4. We see that for x = 0.70, 0.80, 0.90, the empirical ratios 0.99, 1.11, 1.27 are very close to the theoretical
2-earner5 ratios 1.00, 1.15, 1.32. It should be noted that the theory applies to pre-tax incomes while our data is a mixture of post-tax (disposable) and pre-tax incomes. The latter yield empirical ratios even closer to the theoretical ones.
[Table 1]
[Figure 2A]
[Figure 2B]
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Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 1 also brings up a second point of some interest: while the theoretical 1-earner and
2-earner household distributions generally give rise to somewhat different coupling coefficients at any given population proportion, they yield the same constant at x = 0.75.
Given that the actual data encompasses a mixture of 1-earner and 2-earner households,
EPTC theory implies that observed ratios will have a minimum coefficient of variation (CV) at x = 0.75. Decile data can only bracket this population proportion between 0.70 and 0.80, but we will see in the next section that the lowest empirical CV is at x = 0.80. Thus, in addition to being calculable from quintile data, the 80 percent per capita income also has the best fit because it is "immune" to the earner-composition of households. We therefore propose the average income of the bottom 80% of the population (VMI) as an inequality- sensitive measure of economic well-being that best combines information on both changes in the average and in the distribution of national income.
Per capita income and economic well-being
Although, GDP per capita (GDPpc) is by far the most popular measure of international levels of development (Frumkin, 2000, pp. 144-154), even though it is known to be an imperfect proxy for important factors such as health, education and well-being (Cowen, 2007).
Therefore, there have been various attempts to bring the issue of distribution into the picture to end the tyranny of averages represented by GDPpc. One important alternative has been to work directly with the variables of concern, as in the United Nations
Development Programme (UNDP) Human Development Index (HDI) which combines GDPpc with life expectancy and schooling into a single composite index (UNDP, 1990, p. 12). But
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Some Universal Patterns in Income Distribution: An Econophysics Approach
the HDI is difficult to compile and is only available for recent years. Because it is an index, it cannot tell us about the absolute standard of living of the underlying population, which is why GDPpc remains so popular (Hicks, 2004, pp. 2-3). Moreover, the rankings produced by the two measures are quite highly correlated (Kelley, 1991, pp. 322-323). In any case, both measures suffer from the fact that "they are averages that conceal wide disparities in the overall population" (UNDP, 1990). There is widespread agreement that international economic comparisons should not ignore inequality. But there is also considerable debate on how exactly to bring inequality into the picture (Gruen and Klasen, 2008, p. 213).
The traditional literature is posed in terms of social welfare functions. Atkinson (1970) developed a measure of welfare loss which is the equally distributed equivalent income -- i.e. "the amount of income that, if distributed equally, would yield the same welfare as the actual mean income and its present (unequal) distribution". The theoretical foundations for the Atkinson measures can be found in social welfare functions which are "additively separable functions of individual incomes [Yi] … based [in turn] on individualistic utility functions where people only care about their own incomes". In this case an increase in inequality reduces welfare according to the degree of "aversion to inequality factor" ( )
1 N 1− 1 1− in the general measure of welfare loss AY= i which can be directly treated as N i=1 a measure of social welfare. On the other hand, Degum (1990) derives the social welfare
(1 − G) functions Wy= based on the assumption that individuals are negatively affected 1 + G
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Some Universal Patterns in Income Distribution: An Econophysics Approach
not only by the level of income inequality but also by their envy of people ahead on them in the queue.
An alternate approach is available in Sen's (1976) measure Wy= (1 − G) , where y = national per capita income and G = the Gini coefficient. He derives this from an assumed social welfare function in which "the weight of a person’s income depends inversely on the rank in the income distribution. Measures of inequality other than the Gini coefficient can also be used, as in Lambert's general function W = W(x, I) or some convenient particular form such as W=− y(1 I ), where I = some general measure of inequality 01I (Gruen and Klasen, 2008, pp. 214-217).
It is interesting to note that the Sen measure was included in the 1993 Human
Development Report (HDR) of the UNDP as a basis for international comparisons, but then dropped because of concerns about its theoretical properties (Foster, Lopez-Calva and
Szekely, 2005 ; Hicks, 1997 ; 2004 ; UNDP, 1993). In the face of renewed interest in inequality after the Great Recession in 2008, the UNDP introduced a version of the previously mentioned Atkinson measure to calculate an inequality-adjusted HDI in the
2010 Human Development Report (Alkire and Foster, 2010 ; UNDP, 2010). The main difficulty in this case was that there was no justification for any particular inequality aversion factor ( ) . While experimental behavioral economics may be viewed as having
"discovered" that that people of different cultures and backgrounds care about equality and fairness (Engelmann and Strobel, 2004, pp.866-868; Fehr and Schmidt, 1999, pp. 855-
856; Hoffman, McGabe and Smith, 1996, pp. 297-300; World Bank, 2006, Ch 4), none of this
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Some Universal Patterns in Income Distribution: An Econophysics Approach
provides much support for any particular level of (Carlsson, Daruvala and Olof, 2005, p.
376).
The social welfare function approach to international comparisons has been criticized because it requires strong theoretical assumptions about individual behavior and psychology, about appropriate measures of individual “well-being” such as utility, about the aggregation of any such measures into indexes of social welfare, and about the effects of income, inequality, education, etc. on all of this (Fleurbaey and Mongin, 2005). Our own approach is different in two dimensions. First, we have already established that subgroup per capita incomes are particular combinations of the national average and the national degree of inequality, the particularity being that each subgroup measure has its own objective coupling coefficient. Second, we would argue that the per capita income of some large majority subgroup such as bottom 70 or 80 percent has direct political and social significance, which is where the social-subjective plays a role6. The 80 percent coupling coefficient is empirically the most robust across countries and time (Figure 4), which is why we favor the VMI as an inequality-sensitive measure of economic well-being.
Empirical evidence
Our distribution data is from the World Income Inequality Database (WIID3.4) published by the United Nations University (UNU) and the World Institute for Development Economics
Research (WIDER) (UNU-WIDER, 2017 May, V 3.4). Further details are in the Data Appendix.
Figures 2A displays the ratio of the relative per capita income of the bottom 80 percent of the population to (1-G), while Figure 2B displays the same ratio for the bottom 70 percent
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Some Universal Patterns in Income Distribution: An Econophysics Approach
across all countries. These ratios are the coupling coefficients derived in equation (4) and their striking uniformity across countries is an indication of the strength of the theory.
[Figure 3]
Figure 3 displays the values of the coupling coefficient a(x) where x represents the cumulative proportion of the population by decile in 2014, for all values of x in all countries.
We can see from the figure that the coupling coefficient a(x) of the top four deciles (60, 70,
80 and 90) are the most stable across countries. This is apparent in Table 2 where the coefficients of variation (CV) for these four deciles range from 2% to 5%. The 70 and 80 decile coefficients are particularly stable having CVs of only 2%. Furthermore, the empirical coefficients are close to the theoretical values. As previously noted, the differences between empirical and theoretical values are most likely due to the fact that the theory is in terms pre-tax incomes while the data includes both post-tax (disposable) and pre-tax incomes.
[ Table 2]
[Figure 4]
Figure 4 displays the coupling coefficient a(80) for 66 countries from 1990 to 2015, which shows that it is very stable across countries and also over time. A similar pattern exists for a(70). It is on this basis that we propose two universal rules. First, the ‘1.0 rule’ which states that the per capita income of the bottom 70 percent is always equal to inequality- discounted GDP per capita. This is in fact Sen’s measure of welfare, derived in our case from
EPTC theory.
(6) 푦̅(70) = 1.0 (1 − 퐺)푦̅ 14
Some Universal Patterns in Income Distribution: An Econophysics Approach
This rule also allows us to provide a new interpretation of the Gini Coefficient, one which is both simple and intuitive: (1-G) represents the relative disposable per capita income of the first seventy percent of a nation's population; equivalently, G represents the percentage difference between GDPpc and the per capita income of the first 70 percent of the income distribution.
푦̅− ̅푦(70) (7) 퐺 = 푦̅
The second universal is the ‘1.1 rule’ which states that the average income of the bottom
80 percent is equal to 1.1 times inequality-discounted GDPpc.
(8) 푦̅(80) = 1.1 (1 − 퐺)푦̅
A final consideration is that the average income of the bottom 80 percent (VMI) provides a different ranking of countries than traditional GDPpc. Table 3 displays both measures by country, along with a measure of the per capita income of the top twenty percent of the population which we call the Affluent Minority Income (AMI). The table also displays national rankings by GDPpc and VMI per capita, as well as the change in ranking in going from the first to the second measure, countries being ranked according to this change in rank.
Three interesting patterns emerge from this data. First of all, we find a great range in VMI's: at the top end of the scale Luxemburg ($89,580), Norway ($85,287) and the Switzerland
($65,287), and at the bottom end Tajikistan ($782), Burkina Faso ($487) and Niger. ($301).
A second finding is displayed in the last row of Table 3, which shows the per capita incomes of the rich (AMIs) have a considerably lower coefficient of variation (82 percent) than that
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Some Universal Patterns in Income Distribution: An Econophysics Approach
of the vast majority incomes (96 percent). Apparently, the international rich are more alike across nations than are the rest of the world's population. The third finding is that the ratio of VMI to GDPpc (i.e. the IR80) varies substantially across countries, so that country rankings differ according to which measure is used. The last three columns in Table 3 display these ranking numerically for each country in the sample, along with its change in rankings when one uses VMI in place of GDPpc. We see that 24 out of 66 countries gain in ranking
(Belarus, Finland and Ukraine rise 3 places in ranking) while 188 countries lose ranking (the
USA goes down 7 places from 7th to 14th).
[Table 3]
We also performed some experiments which led us to conclude that our two rules are not statistical artifacts. First, we checked the statistical procedures used for fitting Lorenz curves to actual data. Second, we generated the proportionality coefficients associated with the probability distribution functions (pdfs) typically used to study income inequality.
Third, we did the same for the functional forms typically used to represent Lorenz curves.
Our first experiment indicates that our two rules are probably not statistical artifacts of the procedures used to fit Lorenz curves. We generated data from a lognormal pdf with Gini coefficients ranging from 0.10 to 0.50 and found that the widely used World Bank package
POVCAL accurately estimated the Lorenz curves and Gini coefficients. We next investigated the properties of probability distributions themselves, using the three most widely used probability distributions in the study of income inequality: the Pareto, the Exponential and the Lognormal (Crow and Shimizu, 1988, p. 233-237; Dovring, 1991, pp. 30-31; Kleiber and
Kotz, 2003, p. 14; Silva and Yakovenko, 2004). We picked combinations of parameter values 16
Some Universal Patterns in Income Distribution: An Econophysics Approach
for each pdf that gave us Gini coefficients within the 0.30-0.70 range observed in the actual international data, checking to make sure that the corresponding Lorenz curves are viable.
As detailed in the Distribution Theory Appendix, we then calculated the proportionality coefficients ax( ) for population proportions between x = 0.70 and x = 0.90. For all three pdfs at x = 0.80 the coefficient value is not far from our empirically observed number of 1.1: for the Pareto pdf it is about 1.03 over the whole range of Gini's; for the exponential pdf it is constant at essentially 1.2; and for the lognormal pdf it is between 1.10-1.14. Secondly, the population proportion which comes closest to following our 1.0 Rule is roughly x = 0.75 in all three theoretical pdfs, which is not far from x = 0.70 predicted in the EPTC approach and found in the data. Finally, it is striking that the minimum coefficient of variation (CV) for these theoretical ratios occurs at x = 0.80.
Our third line of investigation focused on the functions commonly used to fit Lorenz curves to actual data. The three general functional forms considered were the exponential, the
General Pareto (which subsumes the Ortega, the RGKO, and the classical Pareto), and the
Beta (Chotikapanich and Griffiths, 2003 pp. 7-8). As detailed in the Distribution Theory
Appendix, the exponential functional form gives a coefficient a(0.80) between 1.07-1.19 which is close to observed empirical ratio of 1.11. But for x = 0.70 the spread of a(0.70) is between 0.61-0.99, so that the average is well below our empirical 1.0. The General Pareto is more complex, and the generalized Beta functional form the most complicated of all, but here too for plausible ranges of their parameters they give similar results.
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Some Universal Patterns in Income Distribution: An Econophysics Approach
We would argue that the generally used pdfs and functional forms are "standard" precisely because they provide good fits to an overall income distribution which is itself derived from two different types of underlying distributions whose union imposes particular shape restrictions on the overall Lorenz curve. We believe that this is one of the major contributions of the econophysics "two-class" (EPTC) approach.
Some General Implications
Three main implications can be adduced from our empirical results and theoretical investigations. First, one can usefully supplement traditional rankings in terms of GDPpc with those in terms of the VMI because the latter combines income levels and inequality into a simple measure: the income per capita of the vast majority of the population in each nation. This measure has an intuitive content which need not be tied to that of traditional welfare theory. For our second policy implication, it is useful to note that our empirical rules y( x) a( x)(1 − G) y , in which ax( ) =1.0, 1.1 correspond to x = 0.70,0.80 respectively, implies that the per capita income of the great bulk of the population will vary solely with the "inequality discounted real GDP per capita" yy' (1− G) . This is true regardless of any interaction between distribution and growth. But since the evidence does not indicate any robust linkages between the two (Aghion, Caroli and Garcia-Penalosa, 1999 ; Alderson and Nielsen, 2003), we can say something even stronger: increases in per capita income ( y ) and increases in equality (1-G) have different effects on the per capita income of the vast majority of the population in a given country depending on the initial level of G. Let
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Some Universal Patterns in Income Distribution: An Econophysics Approach
y( x ), y , G denote percentage rates of change of majority per capita income, GDPpc and G,
G respectively, and note that the rate of change of (1-G) is −G . Then the general 1− G proportionality rule implies that
G (9) y() x= −+G y 1− G
Thus, while growth in GDPpc will always raise majority per capita incomes by the same percentage, a given percentage reduction in inequality as measured by the Gini coefficient will have more than a proportional effect in countries with Gini Coefficient above 0.50, and less than a proportional effect in the rest. To put it differently, the partial elasticity of majority capita incomes with respect to GDPpc will always equal one, whereas the absolute value of the elasticity with respect to G will be greater than one for G > 0.50, and less than one for G < 0.50. As shown in Table 3, Gini coefficients for the 2014 range from 0.22 (Iceland) to almost 0.52 (Colombia). Thus, the absolute values of the partial elasticities of per capita incomes with respect to G range from 0.33 to 1.1. If we consider data from previous years we see that the number of countries with elasticities above one is not trivial. For example, the percentage of countries reporting G > 0.50 was 30% in 1994, 22% in 2002, and 12% in
2010 of all countries with available data in each year. Table 3 shows that in 2014 three out of sixty-six countries have Gini's above 0.50, which means that for this group the partial effect of a reduction in inequality on the average income of the bottom 80 percent would be greater than that of an increase in GDPpc.
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Some Universal Patterns in Income Distribution: An Econophysics Approach
EPTC equation (3) says (1−G) =( 1 − G ')( 1 − f ) , where f = the share of property income in total household income, and (1'− G ) is a fixed number. This immediately implies that the index of equality (1 − G) is proportional to the wage and salary share in household income
(1 − f ) (Shaikh 2016). Alternately, the index of inequality G is positive linear function of the share of property income.
(10) G= G' +( 1 − G ') f
Equation (10) is a theoretical relation. But given its robust empirical performance, we may conclude that the Gini coefficient will be sensitive to changes in the distribution between wage and property incomes, but not to the distribution of income within either income class. This proposition has found empirical support both inside and outside the EPTC literature (Checchi and García-Peñalosa, 2008, pp. 4-5; Silva and Yakovenko, 2004, p. 3) .
Summary and Conclusions
In this paper, we began by laying out the econophysics "two class" (EPTC) approach to income distribution, which posits and tests the hypothesis the overall distribution of income is composed of two probability distribution functions (pdfs): the exponential pdf characterizing the distribution of wages and salaries, and the Pareto pdf characterizing the distribution of property income. The EPTC approach derives a simple approximation for the corresponding overall Lorenz curve, which we then use to demonstrate that the per capita income of any bottom fraction (x) of the population will be proportional to "inequality- discounted" national income per capita (GDPpc)∙(1-Gini), the constant of proportionality being solely a function of x.
20
Some Universal Patterns in Income Distribution: An Econophysics Approach
We test this theoretical result against a large sample of countries in the WIID database and show that is extremely powerful across countries and through time. These findings give rise to two universal rules. The 1.1 Rule says that the VMI, the income per capita of the bottom
80 percent of a country's population, can be calculated by multiplying inequality adjusted
GDPpc by 1.1. This allows us to simply and accurately estimate the VMI of any country from two easily available national statistics. For the bottom 70 percent of the population, the corresponding coefficient is 1, which implies that inequality-discounted GDPpc is the same as the per capita income of the bottom 70 percent. This latter result is of some interest, because Sen (1976) used traditional welfare theory to derive a measure of social welfare which equals inequality-adjusted GDPpc. 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. But the obverse need not be true, since we would argue that the per capita income of the great bulk of the population has an intuitive content which need not be tied to traditional welfare theory. Indeed, we arrive at this measure and its properties from an entirely different foundation. Additionally, using the VMI instead of GDPpc gives different international rankings of countries. For example, Finland's VMI in 2014 was 22 percent higher than that of the US, even though its GDPpc was 6 percent lower.
We investigated whether our two rules might be statistical artifacts of data-fitting procedures and conclude that they are not. We do find that roughly similar results can be obtained from the probability distribution functions and curves which are commonly used to fit empirical income distributions and Lorenz curves. But we would argue that these functions and curves are popular precisely because they are able to accommodate the 21
Some Universal Patterns in Income Distribution: An Econophysics Approach
shape restrictions inherent in the two distinct underlying pdfs identified by the EPTC approach.
Three main implications can be drawn from our findings. First, that for international comparisons a measure such as the VMI is preferable to GDP because the former combines income levels and inequality into a simple and intuitively monetary statistic: the per capita of the bottom 80 percent of the population. Such a measure is socially and politically relevant in the same way as are measures of poverty. Second, we show that while growth in average per capita incomes and reductions in inequality both raise the VMI, the latter has a larger effect in countries with Gini coefficient greater than 0.50. Finally, we show that the EPTC approach implies that the Gini coefficient is a simple function of the share of property income in total national income. At practical level, this means that the Gini will be sensitive to changes in distribution between labor and property income, but insensitive to changes in distribution within either class. Since incomes in the bottom 97 percent of the population are dominated by labor income, this result is consistent with the well-known insensitivity of the Gini to distributional changes in the middle range.
22
Some Universal Patterns in Income Distribution: An Econophysics Approach
Data Appendix
We use data from the World Income Inequality Database (WIID3.4) published by the United
Nations University (UNU) and the World Institute for Development Economics Research
(WIDER) (UNU-WIDER, 2017 June, V 3.4). This is an updated and modified compilation of the original WIID V 1.0 (September 2000) and an unpublished update by Deininger and
Squire (D&S, 2004) from the World Bank. Most of the distribution data is for income but in some cases, it is for consumption. The original income unit is generally the households or family, but in about 70 percent of the cases the available data is in terms of equivalent personal income, gross or disposable, by quintile or decile. In creating our present database, we removed the following observations: those which did not report at least quintile data, those which did not cover the entire area or population of a country, those with data quality rankings below 2 (1 being the highest), and those with no corresponding GDPpc data in the
World Bank World Development Indicators (World Bank 2017). We also eliminated less well-defined income definitions (such as inequality data based on surveys of earnings or primary income). The data covers both household and personal and data based on gross income, net income or consumption. It should be noted that all the "personal" income data is actually derived from household income data using an equivalence scale (UNU-WIDER,
2005 June, V 2.0a-a, pp. 17-18). The remaining datasets includes 4,288 income inequality data points covering 169 countries over the period 1970 to 2015. Earlier observations are often quite sparse, and many countries have information for only a few years. Table A1 shows the breakdown of our dataset by unit of analysis (person or household) and by
23
Some Universal Patterns in Income Distribution: An Econophysics Approach
income definition (gross income, net income, income, consumption or monetary income)7.
Most of the data points (77%) are based on net income or consumption both representing some form of post-tax income definition. Only 5% of the data is based with certainty on a pre-tax (gross) income definition.
[Table A1]
Distribution Theory Appendix
We examined the properties of the three most widely used probability distributions in the study of income inequality: the Pareto, the Exponential and the Lognormal. We picked combinations of parameter values for each pdf that gave us Gini coefficients within our empirically observed 0.30-0.70, checking to make sure that the corresponding Lorenz curves are viable. For each pdf, the cumulative population proportion (x) is given by its cumulative distribution function (cdf) for incomes below the given value, while the corresponding cumulative income proportion is calculated either from the appropriate formula or by summing calculated income densities. These are used to calculate the coupling coefficients a(x) for x between 0.70-0.90. For the Pareto distribution x = cdf = 1 –
(Ymin/Y)k, y = 1 – (1-x)1-(1/k) , and G = 1/(2k -1 ), where Ymin = the minimum income > 0, and k = the shape parameter > 0. For this simulation, Ymin was fixed and k was varied to generate Gini's within the chosen range. For the 1-earner Exponential distribution
(Dragulescu and Yakovenko, 2001, 586-587), the Lorenz curve y = x + (1-x)∙ln(1-x) is parameter free and the Gini is fixed at G = 1/2 while for the 2-earner G = 3/8 and the Lorenz curve can be derived from the fact that both y and x are functions of a common third
24
Some Universal Patterns in Income Distribution: An Econophysics Approach
variable (Dragulescu and Yakovenko, 2001, 586-588). Finally, for the Lognormal distribution
2 (ln Y − ) 1 − the population proportion pY()= 2 2 was calculated for income levels (Y) Y2 e between 1 and 500, and then cumulated to get the cumulative population proportion (x(Y)).
For each income level, Y∙p(Y) represents the income proportion, and these were cumulated to generate the corresponding cumulative income proportion (y(Y)). The Gini coefficient
n was then calculated from Brown's Formula (Brown, 1994 ):G =−1 −+ ( xxjj−1)( yyjj−1) j=1
. Table 4 displays the ratios a(x) for 0.70 ≤ x ≤ 0.90, with parameter values chosen to give
0.30 ≤ G ≤ 0.70, along with coefficients of variation displayed in the last column. We see that a(0.80) is about 1.03 over the whole range for the Pareto pdf, is constant at 1.2 for the exponential pdf, and ranges 1.10-1.14 for the lognormal pdf. All three of these ranges are close to our empirically observed ratio of 1.11. Conversely, the population proportion which gives us our 1.0 Rule is roughly 75 percent in all three theoretical pdfs, as opposed to seventy percent in the actual data. Remarkably, the minimum coefficient of variation (CV) for these theoretical ratios occurs at x = 0.80, just as it does in the observed data in Figures
6-7.
[Table 4]
Our next investigation focused on the functions used to fit Lorenz curves to actual data. The three general functional forms considered were the exponential, the General Pareto (which subsumes the Ortega, the RGKO, and the classical Pareto), and the Beta (Chotikapanich and
Griffiths, 2003 pp. 7-8). In the case of the exponential functional form the Gini coefficients
25
Some Universal Patterns in Income Distribution: An Econophysics Approach
corresponding to different parameter values can be directly calculated, but in the other two cases they are calculated via Brown's Formula cited previously. Only parameter values corresponding to viable Lorenz curves are retained. The exponential function form of the
ax e −1 11 Lorenz curve is the single parameter function y = with (1 − G) = − a , and a a e −1 e − 1 for x = .80 this gives a(0.80) between 1.07-1.19 which is close to observed empirical ratio of
1.11, but for x = 0.70 the a(0.70) spread is between 0.61-0.99 so that the average is well below our empirical 1.0. The General Pareto is a considerably more complex three
parameter function of the form , where 0, 1,0 1 . For x = yx= 1−(1−x)
0.80 the a(x) ranges from 1.02-1.28 with an average of 1.18, while for x = 0.70 the corresponding ratio ranges from 0.79-1.04 with an average of 0.90. Finally, the generalized
d b Beta functional form is y =−xax (1−x) , where a > 0, 0 < d ≤ 1, 0 < b ≤ 1. This gives rise to the most complicated set of calculations because viable Lorenz curves turned out to high values of the parameter a with corresponding high values of d. Consequently, with the Beta function we were only able to generate Gini coefficients at the low end of our chosen scale, between 0.30-0.40. Table 5 summarizes the results of these experiments.
[Table 5]
26
Some Universal Patterns in Income Distribution: An Econophysics Approach
Figures and tables
Figure 1: The Lorenz Curve
100%
Cumulative proportion of total income C (z) A
B
0% 80% 100% Cumulative proportion of total population (x)
27
Some Universal Patterns in Income Distribution: An Econophysics Approach
Figure 2A: The 1.0 Rule: Coupling coefficient for bottom 70 percent a(0.70) by country for 2014 ( overall average = 0.99) 1.20
1.00
0.80
0.60
0.40
0.20
0.00
Peru
Niger
Brazil
Serbia
Ireland
Austria
Greece
Croatia Iceland
Finland
Georgia
Sweden
Panama
Belgium
Slovenia
Portugal Vietnam
Moldova
Lithuania
Tajikistan
Colombia
Honduras
Argentina
Kyrgyzstan
ElSalvador
Mauritania
Macedonia
Montenegro
Burkina Faso Burkina
New Zealand New
Czech Republic Czech UnitedKingdom
Dominican Republic Dominican
28
Some Universal Patterns in Income Distribution: An Econophysics Approach
Figure 2B: The 1.1 Rule: Coupling coefficient for bottom 80 percent a(0.80) by country for 2014 ( overall average = 1.11) 1.20 1.10 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10
0.00
Peru
Niger
Brazil
Serbia
Ireland
Austria
Greece
Croatia Iceland
Finland
Georgia
Sweden
Panama
Belgium
Slovenia
Portugal Vietnam
Moldova
Lithuania
Tajikistan
Colombia
Honduras
Argentina
Kyrgyzstan
ElSalvador
Mauritania
Macedonia
Montenegro
Burkina Faso Burkina
New Zealand New
Czech Republic Czech UnitedKingdom
Dominican Republic Dominican
29
Some Universal Patterns in Income Distribution: An Econophysics Approach
Figure 3: Coupling coeficient a(x) for 65 countries in 2014 for all deciles 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.20
0.00
Peru
Niger
Brazil
Serbia
Ireland
Austria
Greece
Croatia Iceland
Finland
Georgia
Sweden
Panama
Belgium
Slovenia
Portugal Vietnam
Moldova
Lithuania
Colombia Tajikistan
Honduras
Argentina
Kyrgyzstan
ElSalvador
Mauritania
Macedonia
Montenegro
Burkina Faso Burkina
New Zealand New
Czech Republic Czech
UnitedKingdom Dominican Republic Dominican
a(10) a(20) a(30) a(40) a(50) a(60) a(70) a(80) a(90)
30
Some Universal Patterns in Income Distribution: An Econophysics Approach
Figure 4: Coupling coefficient a(x) from 1990-2015 (overall average = 1.1) 1.20 Germany 1.15 Finland 1.10 Poland
1.05 Romania United Kingdom 1.00 Norway 0.95 Argentina Costa Rica 0.90 Taiwan 0.85 Hungary
0.80 Honduras Colombia 0.75 Uruguay
0.70 Slovenia
2010 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2011 2012 2013 2014 2015 1990
31
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 1: Theoretical and Actual Coupling Coefficients a(x)
Cumulative Population Econophysics Econophysics Actual
Proportion (x) 1-Earner Households 2-Earner Households International
Data
0.70 0.97 1.01 0.99
0.75 1.08 1.08
0.80 1.20 1.15 1.11
0.85 1.33 1.23
0.90 1.49 1.33 1.27
32
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 2: Estimated Values of the Mean and Deviation of the Coupling Coefficients a(x)
Population percentage Average Coupling Coefficient Variation CV (x) Coefficient (a)
0.10 0.41 26.8%
0.20 0.54 18.0%
0.30 0.64 13.3%
0.40 0.73 9.8%
0.50 0.81 6.9%
0.60 0.90 4.2%
0.70 1.00 1.8%
0.80 1.11 1.8%
0.90 1.25 5.1%
33
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 3: VMI and GDP Per Capita Levels (US$) and Rankings by Country (2014)
US$ Rankings
Country Gini GDPpc VMI AMI GDPpc VMI Difference
United 46.40 52,787 32,992 131,968 7 14 -7 States
Paraguay 51.12 4,480 2,461 12,556 48 54 -6
Colombia 52.63 8,031 4,283 23,022 39 43 -4
Ireland 30.80 52,035 39,611 101,728 8 11 -3
Australia 38.95 67,792 46,819 151,685 4 6 -2
Ecuador 44.53 6,074 3,771 15,286 45 47 -2
New Zealand 33.00 42,890 31,685 87,710 15 17 -2
Poland 31.44 13,781 10,413 27,251 31 33 -2
Bolivia 47.63 2,948 1,769 7,665 56 57 -1
Cameroon 46.54 1,365 826 3,525 62 63 -1
Costa Rica 47.84 10,570 6,179 28,131 37 38 -1
Estonia 35.60 19,030 13,797 39,963 26 27 -1
Honduras 49.81 2,137 1,218 5,813 58 59 -1
Malta 27.70 23,930 18,905 44,032 21 22 -1
Mexico 48.21 10,299 5,896 27,910 38 39 -1
Peru 43.48 6,583 4,226 16,011 43 44 -1
Portugal 34.50 21,619 15,755 45,075 24 25 -1
34
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 3: VMI and GDP Per Capita Levels (US$) and Rankings by Country (2014)
US$ Rankings
Country Gini GDPpc VMI AMI GDPpc VMI Difference
Uruguay 40.83 16,881 11,253 39,392 28 29 -1
Argentina 41.13 12,977 8,653 30,271 34 34 0
Brazil 50.73 12,217 6,772 33,996 35 35 0
Burkina Faso 35.30 699 487 1,549 65 65 0
Cyprus 34.80 27,908 19,850 60,142 20 20 0
Denmark 27.70 61,191 48,494 111,980 5 5 0
Dominican 43.10 6,027 3,808 14,902 46 46 0 Republic
El Salvador 41.15 3,896 2,544 9,302 52 52 0
Georgia 40.09 4,274 2,883 9,842 50 50 0
Germany 30.70 46,531 35,887 89,107 13 13 0
Greece 34.50 21,875 16,105 44,953 23 23 0
Guatemala 47.79 3,453 2,020 9,186 55 55 0
Hungary 28.60 13,614 10,602 25,662 32 32 0
Italy 32.40 35,370 26,837 69,503 18 18 0
Latvia 35.50 15,032 10,842 31,793 30 30 0
Luxembourg 28.70 113,752 89,580 210,441 1 1 0
Mauritania 32.42 1,451 1,088 2,900 61 61 0
Mongolia 32.04 4,385 3,270 8,845 49 49 0
Nicaragua 46.45 1,847 1,102 4,826 60 60 0
35
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 3: VMI and GDP Per Capita Levels (US$) and Rankings by Country (2014)
US$ Rankings
Country Gini GDPpc VMI AMI GDPpc VMI Difference
Niger 33.99 416 301 878 66 66 0
Norway 23.50 102,910 85,287 173,404 2 2 0
Panama 49.87 11,686 6,660 31,790 36 36 0
Spain 34.70 29,210 21,652 59,443 19 19 0
Switzerland 29.50 84,659 65,187 162,545 3 3 0
Tajikistan 30.76 1,040 782 2,073 64 64 0
Austria 27.60 50,505 39,899 92,929 10 9 1
Bulgaria 35.40 7,675 5,564 16,117 41 40 1
Czech 25.10 19,916 16,032 35,451 25 24 1 Republic
France 29.20 42,554 32,660 82,129 16 15 1
Indonesia 37.34 3,621 2,486 8,161 54 53 1
Kyrgyzstan 26.82 1,282 1,009 2,374 63 62 1
Lithuania 35.00 15,713 11,372 33,075 29 28 1
Moldova 26.83 2,244 1,777 4,112 57 56 1
Montenegro 31.93 7,186 5,344 14,556 42 41 1
Slovenia 25.00 23,150 18,983 39,819 22 21 1
United 31.60 42,407 32,230 83,118 17 16 1 Kingdom
Vietnam 37.59 1,908 1,321 4,255 59 58 1
36
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 3: VMI and GDP Per Capita Levels (US$) and Rankings by Country (2014)
US$ Rankings
Country Gini GDPpc VMI AMI GDPpc VMI Difference
Slovakia 26.10 18,192 14,667 32,290 27 26 1
Armenia 31.48 3,844 2,880 7,697 53 51 2
Belgium 25.90 46,510 37,906 80,928 14 12 2
Iceland 22.70 47,810 39,742 80,082 12 10 2
Netherlands 26.20 51,574 41,517 91,803 9 7 2
Serbia 38.60 6,354 4,527 13,661 44 42 2
Sweden 25.40 60,283 49,508 103,386 6 4 2
Croatia 30.20 13,575 10,622 25,385 33 31 2
Macedonia 35.20 5,211 3,850 10,658 47 45 2
Belarus 27.18 7,979 6,316 14,629 40 37 3
Finland 25.60 49,638 40,207 87,363 11 8 3
Ukraine 24.09 4,030 3,291 6,983 51 48 3
37
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 4: Probability Distributions and a(x) coefficients
CV
G 0.30 0.40 0.50 0.60 0.70
Pareto Distribution (Ymin = 2.50)
k 2.17 1.75 1.50 1.33 1.21
a(0.70) 0.973 0.959 0.944 0.928 0.911 0.026
a(0.75) 1.001 0.995 0.986 0.976 0.964 0.015
a(0.80) 1.035 1.038 1.038 1.035 1.030 0.003
a(0.85) 1.076 1.091 1.103 1.111 1.115 0.015
a(0.90) 1.128 1.161 1.191 1.216 1.237 0.036
Exponential Distribution (parameter free)
a(0.70) 0.968 -
a(0.75) 1.076 -
a(0.80) 1.195 -
a(0.85) 1.330 -
a(0.90) 1.488 -
Lognormal Distribution (μ = 1.5)
σ 0.55 0.75 0.96 1.20 1.47
a(0.70) 1.003 0.982 0.950 0.896 0.825 0.072
a(0.75) 1.047 1.048 1.036 1.007 0.951 0.037
a(0.80) 1.097 1.121 1.136 1.133 1.106 0.025
38
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 4: Probability Distributions and a(x) coefficients
CV
G 0.30 0.40 0.50 0.60 0.70
a(0.85) 1.155 1.206 1.250 1.288 1.305 0.079
a(0.90) 1.217 1.306 1.393 1.487 1.577 0.164
39
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 5: Beta Functional Form and a(x) for 0.30 ≤ G ≤ 0.40
x = 0.70 0.75 0.80 0.85 0.90
a b d
0.1 0.9 0.80 1.007 1.013 1.019 1.025 1.031
0.75 1.006 1.018 1.032 1.047 1.064 0.3 0.80 1.003 1.014 1.027 1.041 1.058 0.2 0.75 1.011 1.026 1.040 1.056 1.073 0.6 0.80 1.008 1.022 1.036 1.051 1.068
0.75 1.009 1.030 1.052 1.077 1.106
0.3 0.80 1.004 1.023 1.044 1.068 1.096
0.95 0.992 1.007 1.024 1.045 1.070
0.75 1.018 1.041 1.065 1.090 1.117
0.3 0.6 0.80 1.013 1.035 1.058 1.082 1.109
0.95 1.001 1.020 1.040 1.062 1.087
0.75 1.027 1.049 1.070 1.090 1.111
0.9 0.80 1.023 1.043 1.063 1.083 1.104
0.95 1.011 1.029 1.048 1.066 1.085
0.75 1.014 1.045 1.078 1.114 1.158 0.3 0.80 1.006 1.035 1.065 1.100 1.142 0.4 0.75 1.026 1.059 1.093 1.129 1.168 0.6 0.80 1.019 1.050 1.082 1.117 1.155
40
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table 5: Beta Functional Form and a(x) for 0.30 ≤ G ≤ 0.40
x = 0.70 0.75 0.80 0.85 0.90
a b d
0.75 1.039 1.068 1.098 1.127 1.156 0.9 0.80 1.032 1.061 1.089 1.117 1.145
0.3 0.95 0.984 1.015 1.049 1.090 1.141
0.5 0.6 0.95 1.003 1.038 1.075 1.117 1.164
0.9 0.95 1.021 1.053 1.087 1.120 1.155
0.3 0.95 0.968 1.030 1.100 1.183 1.286
0.75 0.6 0.95 1.005 1.068 1.136 1.210 1.294
0.9 0.95 1.035 1.091 1.147 1.204 1.263
0.3 0.95 0.660 0.708 0.764 0.829 0.910
0.8 0.6 0.95 0.805 0.861 0.922 0.988 1.063
0.9 0.95 0.893 0.946 0.999 1.053 1.108
a(x) Mean 0.988 1.017 1.048 1.082 1.120
a(x) Std. Dev. 0.0779 0.0722 0.0691 0.0698 0.0762
a(x) CV 0.0789 0.0710 0.0659 0.0645 0.0680
41
Some Universal Patterns in Income Distribution: An Econophysics Approach
Table A1: Data breakdown by income definition and unit of analysis (household or person)
Income definition Household Person Total
Income, net 154 2,352 2,506
Consumption 19 791 810
Income 33 672 705
Income, gross 91 80 171
Monetary income, disposable 3 23 26
Monetary income, gross 56 12 68
Monetary income 2 2
Total 356 3,932 4,288
42
Some Universal Patterns in Income Distribution: An Econophysics Approach
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Endnotes
1. The key finding is that "the majority of the population … has a very stable in time exponential ("thermal") distribution of income" which is analogous to the equilibrium distribution of energy in statistical physics following the Boltzmann-Gibbs law of the conservation of energy (Dragulescu and Yakovenko, 2002, pp. 1-2). Yakovenko has pointed out that Gibbs developed his notion of the distribution of particles from his study of social patterns. In this regard econophysics is merely returning the favor. 2. The Gini coefficient itself can be characterized in a variety of ways, but none of them is particularly intuitive (Subramanian, 2004, p. 7) 3. The Gini coefficient of the whole Lorenz curve is G = G'+ f(1- G' ), where G' = the Gini of the exponential portion (Silva and Yakovenko, 2004, p. 5), which yields equation (3) in the text. 4. We thank Victor Yakovenko for pointing out that the "personal-equivalent" units upon which our data is based are actually derived from household data (see the Data Appendix), so that the relevant income distribution is the distribution of household incomes. 5. It can be shown that an equally weighted average of 1-earner and 2-earner households will have a IR/(1-G) ratio close to that of the 2-earner households. 6. Gruen and Klasen (2008, p. 6 footnote 6) mention that utility functions are also judged in terms of their "intuitive appeal". 7. For detailed definitions of these income definitions see the WIID3.4 user guide
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