Comparative analysis of households incomes in the European Union and the United States

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1 Coparative analysis of households incoes in the European Union and the United States Rafał Duczal, Maciej Jagielski, Ryszard Kutner Faculty of Physics, University of Warsaw Hoża 69, PL Warsaw, Poland VII Sypozju Fizyki w Ekonoii i Naukach Społecznych UMCS Lublin aja 2014

2 Presentation plan Motivation Data description Copleentary cuulative distribution function Theoretical approach Results and conclusions

3 Data description We used epirical data fro Eurostat s Survey on Incoe and Living Conditions (EU SILC) for years

4 Data description We used epirical data fro Eurostat s Survey on Incoe and Living Conditions (EU SILC) for years We also used epirical data fro the Internal Revenue Service (IRS), the United States tax agency, for years

5 Data description We used epirical data fro Eurostat s Survey on Incoe and Living Conditions (EU SILC) for years We also used epirical data fro the Internal Revenue Service (IRS), the United States tax agency, for years Therefore, we additionally analysed the effective incoe of billionaires in the European Union and the United States by using the Forbes The World s Billionaires rank.

6 Copleentary cuulative distribution function 1 European Union Year 2010 United States Year FIGURE: Copleentary cuulative distribution function of annual total household gross incoe in the European Union, and annual adjusted gross incoe in the United States for year 2010 (EUSILC UDB 2010 version 1 of March 2012).

7 Theoretical approach By considering incoe (t) as a rando variable, we can adopt the threshold nonlinear Langevin equation: ṁ = A() + C()η(t), (1)

8 Theoretical approach By considering incoe (t) as a rando variable, we can adopt the threshold nonlinear Langevin equation: where η(t) white noise ṁ = A() + C()η(t), (1) We assue that household incoe consists of two coponents. First the deterinistic coponent of incoe arises fro the fact that household incoe can take the for of wages and salaries.

9 Theoretical approach By considering incoe (t) as a rando variable, we can adopt the threshold nonlinear Langevin equation: where η(t) white noise ṁ = A() + C()η(t), (1) We assue that household incoe consists of two coponents. First the deterinistic coponent of incoe arises fro the fact that household incoe can take the for of wages and salaries. Second indeterinistic coponent ay express profits which go to household ainly through investents and capital gains.

10 Theoretical approach By considering incoe (t) as a rando variable, we can adopt the threshold nonlinear Langevin equation: where η(t) white noise ṁ = A() + C()η(t), (1) We assue that household incoe consists of two coponents. First the deterinistic coponent of incoe arises fro the fact that household incoe can take the for of wages and salaries. Second indeterinistic coponent ay express profits which go to household ainly through investents and capital gains. Above nonlinear stochastic dynaics equation is equivalent to Fokker Planck (continuity) equation for the probability distribution function, with equilibriu solution: P() = c B() exp where B() = C 2 ()/2. ( init A( ) B( ) d ), (2)

11 The Yakovenko et al. odel As Yakovenko et al. have already found, the coexistence of additive and ultiplicative stochastic processes is allowed. By assuing that these processes are uncorrelated, we get A() = A 0 + a, B() = B 0 + b 2 = b ( ), (3) where 2 0 = B 0/b.

12 The Yakovenko et al. odel As Yakovenko et al. have already found, the coexistence of additive and ultiplicative stochastic processes is allowed. By assuing that these processes are uncorrelated, we get A() = A 0 + a, B() = B 0 + b 2 = b ( ), (3) where 2 0 = B 0/b. This consideration leads to a significant Yakovenko et al. odel with the probability distribution function given by ( ) exp 0 P() = C T arc tg 0 [ ( ) ] α+1, (4) where T = B0 A 0, α+1 2 = a 2b + 1, C = const and b0 2 0 incoe border between low- and ediu-incoe society class.

13 The Extended Yakovenko et al. odel We assued that the foralis of the incoe change is the sae for the whole society. This foralis is expressed by the threshold nonlinear Langevin equation where particular dynaics distinguishes the range of the high-incoe society class fro those of the others: { A A() = < () = A 0 + a, if < 1, A () = A 0 + (5) a, if 1, B() = B 0 + b 2 = b ( ) (6) where 2 0 = B 0/b and 1 incoe border between ediu- and high-incoe society class.

14 The Extended Yakovenko et al. odel We assued that the foralis of the incoe change is the sae for the whole society. This foralis is expressed by the threshold nonlinear Langevin equation where particular dynaics distinguishes the range of the high-incoe society class fro those of the others: { A A() = < () = A 0 + a, if < 1, A () = A 0 + (5) a, if 1, B() = B 0 + b 2 = b ( ) (6) where 0 2 = B 0/b and 1 incoe border between ediu- and high-incoe society class. We get { c exp( ( 0 /T ) arc tg(/ 0 )) if < [1+(/ P() = 0 ) 2 ] (α+1)/2 1 c exp( ( 0 /T 1 ) arc tg(/ 0 )) if [1+(/ 0 ) 2 ] (α 1 +1)/2 1 where: α = 1 + a/b, α 1 = 1 + a /b, T = B 0 /A 0, T 1 = B 0 /A 0, C = const and b 0 2 C = C exp ( ( 1 T 1 T 1 ) 0 arc tg 1 0 ) [ 1 + ( 1 0 ) 2 ] (α1 α)/2.

15 Epirical results 1 European Union 0.01 Year 2007 T=37000; T=3000 0=160000; 0=20000 Α=2.735; Α= T1=480000; T1= =480000; 1=50000 Α1=0.79; Α1= United States Year T=48430; T=520 0=135000; 0=20000 Α=1.83; Α= T1=450000; T1= =450000; 1=50000 Α1=1.336; Α1= FIGURE: Coparison of the copleentary cuulative distribution function, based on the Extended Yakovenko et al. forula with the EU household incoe epirical data set and US individual incoe data set. The first and the second vertical lines are placed at 0 and 1, respectively (EUSILC UDB 2007 version 3 of March 2010).

16 Epirical results 1 European Union 0.01 Year 2008 T=38000; T=3000 0=120000; 0=20000 Α=2.965; Α= 10 4 T1=450000; T1= =450000; 1=50000 Α1=0.890; Α1= United States Year T=48740; T=520 0=135000; 0=20000 Α=1.85; Α= T1=460000; T1= =460000; 1=50000 Α1=1.381; Α1= FIGURE: Coparison of the copleentary cuulative distribution function, based on the Extended Yakovenko et al. forula with the EU household incoe epirical data set and US individual incoe data set. The first and the second vertical lines are placed at 0 and 1, respectively (EUSILC UDB 2008 version 2 of August 2010).

17 Epirical results 1 European Union 0.01 Year 2009 T=37000; T=3000 0=145000; 0=20000 Α=2.974; Α= 10 4 T1=290000; T1= =290000; 1=50000 Α1=2.608; Α1= United States Year T=48050; T=670 0=135000; 0=20000 Α=1.9; Α= 10 7 T1=500000; T1= =500000; 1=50000 Α1=1.451; Α1= FIGURE: Coparison of the copleentary cuulative distribution function, based on the Extended Yakovenko et al. forula with the EU household incoe epirical data set and US individual incoe data set. The first and the second vertical lines are placed at 0 and 1, respectively (EUSILC UDB 2009 version 3 of March 2012).

18 Epirical results 1 European Union 0.01 Year 2010 T=38000; T=3000 0=135000; 0=20000 Α=3.153; Α= T1=450000; T1= =450000; 1=50000 Α1=0.77; Α1= United States Year T=48680; T=780 0=135000; 0=20000 Α=1.86; Α= T1=420000; T1= =420000; 1=50000 Α1=1.395; Α1= FIGURE: Coparison of the copleentary cuulative distribution function, based on the Extended Yakovenko et al. forula with the EU household incoe epirical data set and US individual incoe data set. The first and the second vertical lines are placed at 0 and 1, respectively (EUSILC UDB 2010 version 1 of March 2012).

19 Epirical results 10 5 United States Year T=48050; T=670 0=130000; 0=20000 Α=1.485; Α= United States Year T=48680; T=780 0=130000; 0=20000 Α=1.395; Α= FIGURE: Coparison of the copleentary cuulative distribution function, based on the Yakovenko et al. forula with the US individual incoe data set. The vertical line is placed at 0.

20 USA Yakovenko odel vs. Extented Ykovenko odel Year Yakovenko odel Extended Yakovenko odel α α α ,3490 1,930 1, ,3508 1,877 1, ,3445 1,833 1, ,3810 1,850 1, ,4846 1,90 1, ,3948 1,865 1,3948 Table: Pareto exponents of the Yakovenko odel and the Extended Yakovenko odel

21 EU vs. USA Siilarities: we can describe incoe of all society classes by the Extended Yakovenko odel

22 EU vs. USA Siilarities: we can describe incoe of all society classes by the Extended Yakovenko odel dynaics of Pareto exponents in the period of the global financial crisis. FIGURE: The variability of the Pareto exponents in years in the European Union and the United States.

23 EU vs. USA Differences: two incoe society classes in the United States vs. three incoe society classes in the European Union

24 EU vs. USA Differences: two incoe society classes in the United States vs. three incoe society classes in the European Union values of Pareto exponents

25 EU vs. USA Differences: two incoe society classes in the United States vs. three incoe society classes in the European Union values of Pareto exponents for the United States we can use both odels the Yakovenko odel and the Extended Yakovenko odel, as well. For the European Union we can use only the Extended Yakovenko odel

26 EU vs. USA Differences: two incoe society classes in the United States vs. three incoe society classes in the European Union values of Pareto exponents for the United States we can use both odels the Yakovenko odel and the Extended Yakovenko odel, as well. For the European Union we can use only the Extended Yakovenko odel incoe borders year area USA UE USA UE USA UE T 0 α $ $ ,930 $ $ ,923 $ $ ,877 $ $ ,959 $ $ ,833 $ $ ,897 1 $ $ $ $ $ $ T 1 $ $ $ $ $ $ α 1 1,3543 0,696 1,3458 0,76 1,3364 0,69 years area USA UE USA UE USA UE T 0 α $ $ ,850 $ $ ,886 $ $ ,90 $ $ ,993 $ $ ,865 $ $ ,128 1 $ $ $ $ $ $ T 1 $ $ $ $ $ $ α 1 1,3810 0,89 1,4513 2,71 1,3948 0,75

27 Conclusions in the United States, in contrast to the European Union, there are only two well-distinguished incoe society classes in both cases (EU and US) Extended Yakovenko odel describes well the incoes of the societies the weak Pareto law exponents are approxiately equal to 3 for the ediu-incoe society class and equal to 1 for the high-incoe society class in the European Union the weak Pareto law exponents are approxiately equal to 1.5 for the ediu- and high-incoe society classes in the United States there ay be a correlation between the volatility of Pareto exponents and the recent financial crisis

28 Publications [1] M. Jagielski, R. Kutner, Extended Yakovenko forula in odelling of incoe distribution in the European Union, Acta Physica Polonica B (2014), in print. [2] M. Jagielski, R. Kutner, Modeling of incoe distribution in the European Union with the Fokker Planck equation, Physica A 392 (2013), pp [3] M. Jagielski, R. Kutner, Ab initio analysis of all incoe society classes in the European Union, Acta Physica Polonica A 123 (2013), pp [4] M. Jagielski, R. Kutner, M. Pęczkowski, Preliinary Coparison of Households Incoe in Poland with European Union and United States Ones by Using the Statistical Physics Methods, Acta Physica Polonica A 121 (2012), pp. B47 B49 [5] M. Jagielski, R. Kutner, Modelowanie zaożności polskich gospodarstw doowych etodai statystycznyi, Ekonoia 25 (2011), pp [6] M. Jagielski, R. Kutner, Study of Households Incoe in Poland by Using the Statistical Physics Approach, Acta Physica Polonica A 117 (2010), pp

29 Thank you for your attention!

30 Prawo Efektów Proporcjonalnych Π() = 1 2 [ ( )] ln µ 1 erf. 2σ Rok 2007 Μ=1950; s Μ = Σ=0.6002; s Σ = Rok 2008 Μ= ; s Μ = Σ=0.6178; s Σ = RYSUNEK: Dopasowanie dopełnienia dystrybuanty rozkładu log-noralnego (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty).

31 Prawo Efektów Proporcjonalnych Π() = 1 2 [ ( )] ln µ 1 erf. 2σ Rok 2009 Μ= ; s Μ = Σ=0.6137; s Σ = Rok 2010 Μ= ; s Μ = Σ=0.6127; s Σ = RYSUNEK: Dopasowanie dopełnienia dystrybuanty rozkładu log-noralnego (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty).

32 Słabe prawo Pareto Π() (/ 0 ) α Rok 2007 Α=2.809; s Α = =24179; s 0 = Rok 2008 Α=3.146; s Α = =30934; s 0 = RYSUNEK: Dopasowanie słabego prawa Pareto (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty). Zielone linie oznaczają przedziały ufności dla zebranych danych epirycznych na pozioie 95%.

33 Słabe prawo Pareto Π() (/ 0 ) α Rok 2009 Α=3.124; s Α = =32810; s 0 = Rok 2010 Α=3.095; s Α = =34395; s 0 = RYSUNEK: Dopasowanie słabego prawa Pareto (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty). Zielone linie oznaczają przedziały ufności dla zebranych danych epirycznych na pozioie 95%.

34 Uogólniony odel Lotka-Volterra α 1 Γ(α, x ) Π(x) = 1. Γ(α) x x Rok 2007 Α=3.572; s Α = x 10 4 Rok 2008 Α=3.546; s Α = x RYSUNEK: Dopasowanie uogólnionego odelu Lotka-Volterra (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty). Zielone linie oznaczają przedziały ufności dla zebranych danych epirycznych na pozioie 95%.

35 Uogólniony odel Lotka-Volterra α 1 Γ(α, x ) Π(x) = 1. Γ(α) x Rok 2009 Α=3.602; s Α = x x Rok 2010 Α=3.558; s Α = x RYSUNEK: Dopasowanie uogólnionego odelu Lotka-Volterra (linia ciągła) do epirycznego dopełnienia dystrybuanty rocznych dochodów rozporządzalnych gospodarstw doowych w Polsce (punkty). Zielone linie oznaczają przedziały ufności dla zebranych danych epirycznych na pozioie 95%.

36 Rankingi gospodarstw doowych o wysokich dochodach Bogactwo Rok 2007 Α ranking =0.89; s Αranking =0.02 Const=23.50; s Const = Α Pareto =1/Α ranking =1.12; s ΑPareto = Ranga l Bogactwo Rok 2008 Α ranking =0.88; s Αranking =0.02 Const=23.62; s Const = Α Pareto =1/Α ranking =1.13; s ΑPareto = Ranga l RYSUNEK: Ranking najbogatszych Polaków (linia ciągła oznacza dopasowanie, a punkty dane epiryczne)

37 Rankingi gospodarstw doowych o wysokich dochodach Bogactwo Rok 2009 Α ranking =0.83; s Αranking =0.02 Const=23.12; s Const = Α Pareto =1/Α ranking =1.20; s ΑPareto = Ranga l Bogactwo Rok 2010 Α ranking =0.89; s Αranking =0.02 Const=23.24; s Const = Α Pareto =1/Α ranking =1.12; s ΑPareto = Ranga l RYSUNEK: Ranking najbogatszych Polaków (linia ciągła oznacza dopasowanie, a punkty dane epiryczne)

38 Podsuowanie Tabela: Modele opisujące dochody gospodarstw doowych w Polsce i Unii Europejskiej. Gospodarstwa Polska Unia doowe Europejska o niskich Prawo Efektów Proporcjonalnych, Prawo Boltzanna-Gibbsa, dochodach uogólniony odel Lotka-Volterra rozszerzony odel Yakovenko o średnich Słabe prawo Pareto, Słabe prawo Pareto, dochodach uogólniony odel Lotka-Volterra rozszerzony odel Yakovenko o wysokich Słabe prawo Pareto Słabe prawo Pareto, dochodach rozszerzony odel Yakovenko

39 Podsuowanie Wyznaczone paraetry pochodzące z różnych odeli ogą, jak przypuszcza, stanowić wskaźniki, czy nawet prekursory kryzysu. Kryzys nie wpływa na gospodarstwa doowe o niskich dochodach (paraetry dopełnienia dystrybuanty rozkładu log noralnego praktycznie nie zieniają się), natoiast prowadzi do niejszego rozwarstwienia społecznego wśród gospodarstw doowych o średnich i wysokich dochodach (oba wykładniki Pareto rosną). Warto jednak zauważyć, że w przypadku gospodarstw doowych o wysokich dochodach wahania wykładnika Pareto są stosunkowo niewielkie co oznacza, że skutki finansowe kryzysu dla najbogatszych obywateli społeczeństwa polskiego nie były aż tak dotkliwe, jak w innych krajach. Jednak w celu wyprowadzenia dalszych wniosków niezbędne są tutaj dodatkowe badania, polegające przede wszystki na przeprowadzeniu porównań z wcześniejszyi kryzysai.