Income Distribution in the European Union Versus in the United States
aa r X i v : . [ q -f i n . GN ] N ov Income Distribution in the European Union Versus in the UnitedStates
Maciej Jagielski ∗ , Rafa l Duczmal, Ryszard Kutner Faculty of Physics, University of Warsaw, Pasteura 5, PL-02093 Warszawa, Poland
Abstract
We prove that the refined approach – our extension of the Yakovenko et al. formalism –is universal in the sense that it describes well both household incomes in the EuropeanUnion and the individual incomes in the United States for social classes of any income.This formalism allowed the study of the impact of the recent world-wide financial crisis onthe annual incomes of different social classes. Hence, we indicate the existence of a possibleprecursor of a market crisis. Besides, we find the most painful impact of the crisis on incomesof all social classes.
Keywords:
Income distribution, Yakovenko model, financial crisis
1. Introduction
One of the major trends having a long history in socio- and econophysics is the study ofincome and wealth redistribution in society and the analysis of social inequalities. Severalmodels trying to explain the microscopic mechanisms of income dynamics of individuals orhouseholds were proposed [1–11].However, (to the best of our knowledge) none of the models that have been developed sofar give an analytic description of the annual household or individual incomes of all socialclasses (i.e. the low-, medium-, and high-income classes) by a single formula based on aunified formalism. Recently [12], we extended the Yakovenko et al. model providing, indeed,such a unified formalism.In the present paper we show that the formula which we derived within this unified for-malism, containing a low number of free parameters, satisfactorily reproduces the empiricalcomplementary cumulative distribution functions (CCDFs) both for the European Union(EU) and for the United States (US). The cumulative distribution function is the mainstatistical tool used in this context, that is, the descriptive statistics technique is involvedherein to analyse data. ∗ Corresponding author. Tel.: +48 22 5532730; fax: +48 22 55 32 999.
Email address: [email protected] (Maciej Jagielski) The complementary cumulative distribution function is the probability that the independent stochasticvariable takes a value larger than some fixed one.
Preprint submitted to Physica A November 7, 2014 . Comments on formula
To describe the income of all social classes in the US and the EU, we used our extendedYakovenko et al. formalism (EYF) [12, 13].As for the Yakovenko et al. model, the coexistence of multiplicative and additive processeson the level of the Langevin equation and hence the Fokker-Planck one, is also allowed forthe EYF. That is, we assume that household or individual incomes are determined by: (i)systematic wages and salaries and/or (ii) random profits that go to households or individualsmainly through financial investments and/or capital gains. Furthermore, for the EYF weassume that the formalism of the income change is the same for the entire society, however,its detailed dynamics distinguishes well the ranges of individual income social classes, inparticular, of the high-income social class from that of others (see [12] for details).We found, in the framework of the EYF, the equilibrium probability distribution functionin the form [12, 13] P eq ( m ) ∝ ( exp( − ( m /T ) arctan( m/m ))[1+( m/m ) ] ( α +1) / , if m < m − ( m /T ) arctan( m/m ))[1+( m/m ) ] ( α / , if m ≥ m (1)where parameter m is a crossover (border) income between the low- and medium-incomesociety classes, while parameter m is an analogous border income but between the medium-and high-income social classes. Parameter T can be interpreted as an average income perhousehold or individual within the low- and medium-income social classes, while interpreta-tion of parameter T is given further in the text. The shape parameters α and α are thePareto exponents, describing the income inequality within the medium- and high-incomesociety classes, respectively. The CCDF considered below is, indeed, an integrated quantityof the above given distribution function.
3. Remarks on databases
In the case of the European Union we exploit the empirical data from Eurostat’s Surveyon Income and Living Conditions (EU-SILC) [14–19] for the years 2005-2010. This databasecontains information on the demographic characteristics of households, their living condi-tions, as well income as economic activity. In our analysis we chose the total household grossincome variable. However, Eurostat’s EU-SILC database contains only a few observationsconcerning the income of households belonging to the high-income social class, which is in-sufficient to subject to any statistical description. In order to improve the statistics for thehigh-income social class, we additionally analysed the effective income of billionaires in theEU by using the Forbes ranking ‘The World’s Billionaires’ [20] (see [12, 13] for more details).In the case of the United States we used the empirical data from the Internal RevenueService (IRS), the US government tax agency, for the years 2005-2010 [21]. We chose the The term billionaire used herein is equivalent (as in the US terminology) to the term multimillionaire used in the European terminology. djusted gross income variable as the only one accessible in the context of our comparativeanalysis. Similarly, as for the EU-SILC, the IRS database does not contain observations onthe individuals belonging to the high-income social class. Again, in order to consider thehigh-income social class, we additionally analysed the effective income of billionaires in theUS by using the same Forbes ranking as mentioned above.By using the EU-SILC database as well as the rank of the richest Europeans and theIRS dataset and a ranking of the richest Americans, we were able to consider incomes of allsocial classes thanks to the joint procedure presented in details in Refs. [12, 13]. Thus, weobtained a data record sufficiently large for the statistical consideration of all social classes.
4. Results and discussion
We compared the theoretical CCDF, based on the probability distribution function P eq ( m ) given by Eq. (1), with: (i) the empirical CCDF of the annual total gross income ofhouseholds in the EU and the corresponding (ii) empirical CCDF of the annual adjusted grossincome of individuals in the US. In our studies we analysed the empirical CCDF constructedby using the well-known Weibull rank formula [22, 23].The two resulting plots, each consisting of the theoretical (solid curves) and empirical(small circles) CCDFs for the EU (the upper curves) and the US (the lower curves), arepresented in a log-log scale in Figs. 1 and 2 for a typical year, 2007, and an exceptionalyear, 2009. Apparently, the EYF describes both the EU and the US empirical CCDFs well.Hence, we were able to provide estimates of the EYF parameters for the years 2005-2010,both for European Union households and United States individuals (cf. Tables 1 and 2).Notably, fits were the best for T = m , which also gives the interpretation of parameter T . Figure 1: A typical comparison of the theoretical CCDFs (solid curves) with the EU household incomeempirical data set (dots – the upper curve) and the US individual income data set (dots – the lower curve),for instance for the year 2007. Notably, the US (theoretical and empirical) curves were shifted down byabout one and a half decade for better distinguishing – their original location is shown in the miniature plotcontaining only the theoretical CCDFs. The solid and dashed pairs of vertical lines concern the EU and theUS curves, respectively. For both pairs the first vertical line is placed at m , while the second one is at m .Apparently, the medium-income social class is much more distinctly formed for the EU than for the US. Remarkably, the values of borders m and m are systematically larger for the EU thanfor the US, except for 2009 (this meaningful exception is discussed further in the text). The3 igure 2: A comparison of the theoretical CCDFs (solid curves) with the EU household income empiricaldata set (dots – the upper curve) and the US individual income data set (dots – the lower curve) for theexceptional year 2009. For better distinguishing, the US curves were shifted down by about two and a halfdecade – their original location is shown in the miniature plot containing only the theoretical CCDFs. Thesolid and dashed pairs of vertical lines play the same role as in Fig. 1. It is striking that the high-incomesocial class almost disappeared in the EU in comparison with the US – for better verification see Fig. 1which also shows how stable the US curve is.Table 1: Parameters m , T , and T (= m ) obtained, in US dollars, for the years 2005-2010, from thecomparison of the theoretical CCDF with the corresponding empirical ones concerning the annual: (i) totalgross income of households in the EU and (ii) adjusted gross income of individuals in the US. The error barsof the parameters do not exceed 18%. European UnitedUnion StatesYear m T T m T T in the EU with individual incomes in the US.Apparently, the range of a medium-income social class (equal to m − m ) is reduced(typically by about 15%) in the case of the US in comparison with the EU (cf. Fig. 1). Thisis a persistent result except for the year 2009, i.e. valid for almost every considered year (inour case from 2005 to 2010). The medium-income social class is more distinct in the EUthan in the US mainly because the difference CCDF( m )-CCDF( m ) is greater for the EU(by a factor of about 1.5). This estimation is also confirmed by the slopes of CCDFs for theEU and the US – the ratio of both slopes again gives a value equal to about 1.5 – for theverification, the corresponding Pareto exponents (given in Table 2) can be compared (see In average, there are about 1.5 employers per single EU household. able 2: Exponents α and α obtained for the years 2005-2010 from the comparison of the theoretical CCDFwith: (i) the empirical CCDF of the annual total gross income of households in the EU, and (ii) the empiricalCCDF of the annual adjusted gross income of individuals in the US. The error bars of exponents do notexceed 4%. European UnitedUnion StatesYear α α α α .
907 0 .
795 1 .
93 1 . .
892 0 .
86 1 .
88 1 . .
735 0 .
79 1 .
83 1 . .
965 0 .
890 1 .
85 1 . .
974 2 .
608 1 .
90 1 . .
153 0 .
77 1 .
86 1 . m increased at the very beginning of the recent world-wide financialcrisis by about 10% – in 2006 in the US and in 2007 in the EU (cf. Table 1) – increasing,thereby, the ranges of the corresponding low-income social classes. Although later theseborders returned to their typical values, this effect could be identified as a clearly inter-pretable possible early-warning signal preceding the crisis – however, to say something moredefinitive, a comparative systematic study concerning all other crises is required.Although the border m in the EU during the exceptional year 2009 was decreased bya factor of about 1.7 in comparison with its typical value (see Table 1), the range of thehigh-income social class decreased. This is because the upper limit of this class drasticallydecreased by more than one decade (compare the upper curves in Figs. 1 and 2). Further-more, since the border m practically did not change (cf. Table 1), the resulting distinctshrink of the medium-income social class is observed.These observations directly relate to the most striking observation that the high-incomesocial class is driven, in the EU, by exponent α which during 2009 almost equals α (upto about 10% accuracy, see Table 2). This observation leads to a situation where the high-income social class plays the role of the medium-income one (see Fig. 2 for details). Hence,although exponent α returned later on to its typical value, this result could be identifiedas a particularly drastic impact of the crisis. That is, the high-incomes social class almostdisappeared in the EU (cf. Fig. 2). Even though for the same year in the US we alsoobserved a slight increase of exponent α , its values are still lower than α (by about 25%).Hence, the shape of the CCDF is quite stable for the US for the years 2005-2010, in spiteof a remarkable increase of the range of the medium-incomes social class (by about 10%) in2009.
5. Conclusions
In the present paper we demonstrated (to best of our knowledge) the first comparisonof incomes in the EU and the US done in such a systematic way. It was possible because5e applied the extended Yakovenko et al. formalism. We proved, herein, that the EYFdescribes the income of the EU households and the US individuals well. By using the EYFwe show that both in the EU and in the US we deal with three income social classes, wherethe medium-income social class has only an intermediate character – one can even say thatboth in the US and in the exceptional year 2009 in the EU, it has a residual character.We found that in the year 2009 the high-income social class abruptly took the role of themedium-income one. This means that the high-income social class, in practice, vanished.That is, the EU society as a whole became poorer than during other years – this is the mostdrastic impact of the crash. In contrast, in 2009 in the US, only a small increase of themedium-income social class was observed, making the range of the high-income social classshorter. Thus, we show that the crisis in the EU has a relatively more painful character thanin the US. During the next year the situation returned to its typical state.Furthermore, an abrupt increase of the upper border of the low-income social class (inthe year 2006 in the US and one year later in the EU) can be considered as an early-warningsignal before the crisis. Nevertheless, the low-income social class in the EU is very similar tothe corresponding one in the US – the shape of both CCDFs is quite stable. The crisis wasmore painful for the medium- and high-income social classes than for the low-income one.We can conclude that the complementary cumulative distribution function, although be-ing a global (macroeconomic) characteristic, is sufficiently sensitive to the crises and crashes,clearly responding over the extended Yakovenko et al. formalism to the income situation ineach income social class, at least in the EU and the US.
Acknowledgements
We thank Victor M. Yakovenko for his helpful discussion.
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