Evidence for the exponential distribution of income in the USA

Transcription

1 Eu. Phys. J. B 2, (21) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Spinge-Velag 21 Evidence fo the exponential distibution of income in the USA A. Dăgulescu and V.M. Yakovenko a Depatment of Physics, Univesity of Mayland, College Pak, MD , USA Received 21 August 2 Abstact. Using tax and census data, we demonstate that the distibution of individual income in the USA is exponential. Ou calculated Loenz cuve without fitting paametes and Gini coefficient 1/2 agee well with the data. Fom the individual income distibution, we deive the distibution function of income fo families with two eanes and show that it also agees well with the data. The family data fo the peiod fit the Loenz cuve and Gini coefficient 3/8 =.375 calculated fo two-eanes families. PACS Ge Dynamics of social systems n Othe topics in aeas of applied and intedisciplinay physics theoy, stochastic pocesses, and statistics 1 Intoduction The study of income distibution has a long histoy. Paeto [1] poposed in 1897 that income distibution obeys a univesal powe law valid fo all times and counties. Subsequent studies have often disputed this conjectue. In 1935, Shias [2] concluded: Thee is indeed no Paeto Law. It is time it should be entiely discaded in studies on distibution. Mandelbot [3] poposed a weak Paeto law applicable only asymptotically to the high incomes. In such a fom, Paeto s poposal is useless fo descibing the geat majoity of the population. Many othe distibutions of income wee poposed: Levy, log-nomal, Champenowne, Gamma, and two othe foms by Paeto himself (see a systematic suvey in the Wold Bank eseach publication [4]). Theoetical justifications fo these poposals fom two schools: socioeconomic and statistical. The fome appeals to economic, political, and demogaphic factos to explain the distibution of income (e.g. [5]), wheeas the latte invokes stochastic pocesses. Gibat [6] poposed in 1931 that income is govened by a multiplicative andom pocess, which esults in a log-nomal distibution (see also [7]). Howeve, Kalecki [8] pointed out that the width of this distibution is not stationay, but inceases in time. Levy and Solomon [9] poposed a cut-off at lowe incomes, which stabilizes the distibution to a powe law. In this pape, we popose that the distibution of individual income is given by an exponential function. This conjectue is inspied by ou pevious wok [1], whee we agued that the pobability distibution of money in a closed system of agents is given by the exponential Boltzmann-Gibbs function, in analogy with the distibution of enegy in statistical physics. In Section 2, a yakovenk@physics.umd.edu we compae ou poposal with the census and tax data fo individual income in USA. In Section 3, we deive the distibution function of income fo families with two eanes and compae it with the census data. The good ageement we found is discussed in Section 4. Speculations on the possible oigins of the exponential distibution of income ae given in Section 5. 2 Distibution of individual income We denote income by the lette (fo evenue ). The pobability distibution function of income, P (), (called the pobability density in book [4]) is defined so that the faction of individuals with income between and +d is P ()d. This function is nomalized to unity (): P ()d = 1. We popose that the pobability distibution of individual income is exponential: P 1 () = exp( /R)/R, (1) whee the subscipt 1 indicates individuals. Function (1) contains one paamete R, equal to the aveage income: P 1 ()d = R, and analogous to tempeatue in the Boltzmann-Gibbs distibution [1]. Fom the Suvey of Income and Pogam Paticipation (SIPP) [11], we downloaded the vaiable TPTOINC (total income of a peson fo a month) fo the fist wave (a fou-month peiod) in Then we eliminated the enties with zeo income, gouped the emaining enties into bins of the size 1/3 k$, counted the numbes of enties inside each bin, and nomalized to the total numbe of enties. The esults ae shown as the histogam in Figue 1, whee the hoizontal scale has been multiplied by 12 to convet monthly income to an annual figue. The solid line epesents a fit to the exponential function (1). In the inset, plot A shows the same data with the logaithmic vetical scale. The data fall onto a staight line,

2 586 The Euopean Physical Jounal B B A. Cumulative pobability Individual annual income, k$ Cumulative pecent of tax etuns A B Cumulative pobability Adjusted goss income, k$ 1 % Individual annual income, k$ Fig. 1. Histogam: distibution of individual income fom the US. Census data fo 1996 [11]. Solid line: Fit to the exponential law. Inset plot A: The same with the logaithmic vetical scale. Inset plot B: Cumulative pobability distibution of individual income fom PSID fo 1992 [12] Adjusted goss income, k$ Fig. 2. Points: Cumulative faction of tax etuns vs. income fom the IRS data fo 1997 [13]. Solid line: Fit to the exponential law. Inset plot A: The same with the logaithmic vetical scale. Inset plot B: distibution of individual income fom the IRS data fo 1993 [14]. whose slope gives the paamete R in equation (1). The exponential law is also often witten with the bases 2 and 1: P 1 () 2 /R2 1 /R1. The paametes R, R 2 and R 1 ae given in line (c) of Table 1. Plot B in the inset of Figue 1 shows the data fom the Panel Study of Income Dynamics (PSID) conducted by the Institute fo Social Reseach of the Univesity of Michigan [12]. We downloaded the vaiable V3821 Total 1992 labo income fo individuals fom the Final Release 1993 and pocessed the data in a simila manne. Shown is the cumulative pobability distibution of income N() (called the pobability distibution in book [4]). It is defined as N() = P ( )d and gives the faction of individuals with income geate than. Fo the exponential distibution (1), the cumulative distibution is also exponential: N 1 () = P 1 ( )d = exp( /R). Thus, R 2 is the median income; of population have income geate than R 1 and only geate than 2R 1.The points in the inset fall onto a staight line in the logaithmic scale. The slope is given in line (a) of Table 1. Table 1. Paametes R, R 2,andR 1 obtained by fitting data fom diffeent souces to the exponential law (1) with the bases e, 2, and 1, and the sizes of the statistical data sets. Souce Yea R ($) R 2 ($) R 1 ($) Set size a PSID [12] ,844 13,62 43, b IRS [14] ,686 13,645 45, c SIPP p [11] ,286 14,61 46, d SIPP f [11] ,242 16,11 53, e IRS [13] ,2 24,399 81, The points in Figue 2 show the cumulative distibution of tax etuns vs. income in 1997 fom column 1 of Table 1.1 of efeence [13]. (We meged 1 k$ bins into 5 k$ bins in the inteval 1 2 k$.) The solid line is a fit to the exponential law. Plot A in the inset of Figue 2 shows the same data with the logaithmic vetical scale. The slope is given in line (e) of Table 1. Plot B in the inset of Figue 2 shows the distibution of individual income fom tax etuns in 1993 [14]. The logaithmic slope is given in line (b) of Table 1. While Figues 1 and 2 clealy demonstate the fit of income distibution to the exponential fom, they have the following dawback. Thei hoizontal axes extend to +, so the high-income data ae left outside of the plots. The standad way to epesent the full ange of data is the so-called Loenz cuve (fo an intoduction to the Loenz cuve and Gini coefficient, see book [4]). The hoizontal axis of the Loenz cuve, x(), epesents the cumulative faction of population with income below, and the vetical axis y() epesents the faction of income this population accounts fo: x() = P ( )d, y() = P ( )d P ( )d (2) As changes fom to, x and y change fom to 1, and equation (2) paametically defines a cuve in the (x, y)- space. Substituting equation (1) into equation (2), we find x( ) =1 exp( ), y( ) =x( ) exp( ), (3) whee = /R. Excluding, we find the explicit fom of the Loenz cuve fo the exponential distibution: y = x +(1 x)ln(1 x). (4)

3 A. Dăgulescu and V.M. Yakovenko: Evidence fo the exponential distibution of income in the USA 587 Cumulative pecent of income Gini coefficient Yea 7% 5% 3% % Family annual income, k$ Cumulative pecent of tax etuns Fig. 3. Solid cuve: Loenz plot fo the exponential distibution. Points: IRS data fo [15]. Inset points: Gini coefficient data fom IRS [15]. Inset line: The calculated value 1/2 of the Gini coefficient fo the exponential distibution. % Family annual income, k$ Fig. 4. Histogam: distibution of income fo families with two adults in 1996 [11]. Solid line: Fit to equation (5). Inset histogam: distibution of income fo all families in 1996 [11]. Inset solid line:.45p 1()+.55P 2(). R dops out, so equation (4) has no fitting paametes. The function (4) is shown as the solid cuve in Figue 3. The staight diagonal line epesents the Loenz cuve in the case whee all population has equal income. Inequality of income distibution is measued by the Gini coefficient G, the atio of the aea between the diagonal and the Loenz cuve to the aea of the tiangle beneath the diagonal: G = 2 1 (x y)dx. The Gini coefficient is confined between (no inequality) and 1 (exteme inequality). By substituting equation (4) into the integal, we find the Gini coefficient fo the exponential distibution: G 1 =1/2. The points in Figue 3 epesent the tax data duing fom efeence [15]. With the pogess of time, the Loenz points shifted downwad and the Gini coefficient inceased fom.47 to.56, which indicates inceasing inequality duing this peiod. Howeve, oveall the Gini coefficient is close to the value.5 calculated fo the exponential distibution, as shown in the inset of Figue 3. 3 Income distibution fo two-eanes families Now let us discuss the distibution of income fo families with two eanes. The family income is the sum of two individual incomes: = Thus, the pobability distibution of the family income is given by the convolution of the individual pobability distibutions [16]. If the latte ae given by the exponential function (1), the two-eanes pobability distibution function P 2 () is P 2 () = P 1 ( )P 1 ( )d = R 2 e /R. (5) The function P 2 () (5) diffes fom the function P 1 () (1) by the pefacto /R, which eflects the phase space available to compose a given total income out of two individual ones. It is shown as the solid cuve in Figue 4. Unlike P 1 (), which has a maximum at zeo income, P 2 () hasa maximum at = R and looks qualitatively simila to the family income distibution cuves in liteatue [5]. Fom the same 1996 SIPP that we used in Section 2 [11], we downloaded the vaiable TFTOTINC (the total family income fo a month), which we then multiplied by 12 to get annual income. Using the numbe of family membes (the vaiable EFNP) and the numbe of childen unde 18 (the vaiable RFNKIDS), we selected the families with two adults. Thei distibution of family income is shown by the histogam in Figue 4. The fit to the function (5), shown by the solid line, gives the paamete R listed in line (d) of Table 1. The families with two adults and moe than two adults constitute 4 and 1 of all families in the studied set of data. The emaining 45% ae the families with one adult. Assuming that these two classes of families have two and one eanes, we expect the income distibution fo all families to be given by the supeposition of equations (1) and (5):.45P 1 ()+.55P 2 (). It is shown by the solid line in the inset of Figue 4 (with R fom line (d) of Tab. 1) with the all families data histogam. By substituting equation (5) into equation (2), we calculate the Loenz cuve fo two-eanes families: x( ) =1 (1 + )e, y( ) =x( ) 2 e /2. (6) It is shown by the solid cuve in Figue 5. Given that x y = 2 exp( )/2 anddx = exp( )d, the Gini

4 588 The Euopean Physical Jounal B Cumulative pecent of family income Gini coefficient Yea Cumulative pecent of families Fig. 5. Solid cuve: Loenz plot (6) fo distibution (5). Points: Census data fo families, [17]. Inset points: Gini coefficient data fo families fom Census [17]. Inset line: The calculated value 3/8 of the Gini coefficient fo distibution (5). coefficient fo two-eanes families is: G 2 = 2 1 (x y)dx = 3 exp( 2 )d =3/8 =.375. The points in Figue 5 show the Loenz data and Gini coefficient fo family income duing fom Table 1 of efeence [17]. The Gini coefficient is vey close to the calculated value Discussion Figues 1 and 2 demonstate that the exponential law (1) fits the individual income distibution vey well. The Loenz data fo the individual income follow equation (4) without fitting paametes, and the Gini coefficient is close to the calculated value.5 (Fig. 3). The distibutions of the individual and family income diffe qualitatively. The fome monotonically inceases towad the low end and has a maximum at zeo income (Fig. 1). The latte, typically being a sum of two individual incomes, has a maximum at a finite income and vanishes at zeo (Fig. 4). Thus, the inequality of the family income distibution is smalle. The Loenz data fo families follow the diffeent equation (6), again without fitting paametes, and the Gini coefficient is close to the smalle calculated value.375 (Fig. 5). Despite diffeent definitions of income by diffeent agencies, the paametes extacted fom the fits (Tab. 1) ae consistent, except fo line (e). The qualitative diffeence between the individual and family income distibutions was emphasized in efeence [14], which split up joint tax etuns of families into individual incomes and combined sepaately filed tax etuns of maied couples into family incomes. Howeve, efeences [13] and [15] counted only individual tax etuns, which also include joint tax etuns. Since only a faction of families file jointly, we assume that the latte contibution is small enough not to distot the tax etuns distibution fom the individual income distibution significantly. Similaly, the definition of a family fo the data shown in the inset of Figue 4 includes single adults and one-adult families with childen, which constitute 35% and of all families. The fome categoy is excluded fom the definition of a family fo the data [17] shown in Figue 5, but the latte is included. Because the latte contibution is elatively small, we expect the family data in Figue 5 to appoximately epesent the two-eanes distibution (5). Technically, even fo the families with two (o moe) adults shown in Figue 4, we do not know the exact numbe of eanes. With all these complications, one should not expect pefect accuacy fo ou fits. Thee ae deviations aound zeo income in Figues 1, 2, and 4. The fits could be impoved thee by multiplying the exponential function by a polynomial. Howeve, the data may not be accuate at the low end because of undeepoting. Fo example, filing a tax etun is not equied fo incomes below a cetain theshold, which anged in 1999 fom $2,75 to $14,4 [18]. As the Loenz cuves in Figues 3 and 5 show, thee ae also deviations at the high end, possibly whee Paeto s powe law is supposed to wok. Nevetheless, the exponential law gives an oveall good desciption of income distibution fo the geat majoity of the population. 5 Possible oigins of exponential distibution The exponential Boltzmann-Gibbs distibution natually applies to the quantities that obey a consevation law, such as enegy o money [1]. Howeve, thee is no fundamental eason why the sum of incomes (unlike the sum of money) must be conseved. Indeed, income is a tem in the time deivative of one s money balance (the othe tem is spending). Maybe incomes obey an appoximate consevation law, o somehow the distibution of income is simply popotional to the distibution of money, which is exponential [1]. Anothe explanation involves hieachy. Goups of people have leades, which have leades of a highe ode, and so on. The numbe of people deceases geometically (exponentially) with the hieachical level. If individual income inceases linealy with the hieachical level, then the income distibution is exponential. Howeve, if income inceases multiplicatively, then the distibution follows a powe law [19]. Fo modeate incomes below $1,, the linea incease may be moe ealistic. A simila scenaio is the Benoulli tials [16], whee individuals have a constant pobability of inceasing thei income by a fixed amount. We ae gateful to D. Jodan, M. Webe, and T. Petska fo sending us the data fom efeences [13,14], and [15], to T. Canshaw fo discussion of income distibution in Bitain, and to M. Gubud fo poofeading of the manuscipt.

5 Refeences A. Dăgulescu and V.M. Yakovenko: Evidence fo the exponential distibution of income in the USA V. Paeto, Cous d Économie Politique (Lausanne, 1897). 2. G.F. Shias, Economic Jounal 45, 663 (1935). 3. B. Mandelbot, Int. Economic Rev. 1, 79 (196). 4. N. Kakwani, Income Inequality and Povety (Oxfod Univesity Pess, Oxfod, 198). 5. F. Levy, Science 236, 923 (1987). 6. R. Gibat, Les Inégalités Économique (Siely, Pais, 1931). 7. E.W. Montoll, M.F. Shlesinge, J. Stat. Phys. 32, 29 (1983). 8. M. Kalecki, Econometica 13, 161 (1945). 9. M. Levy, S. Solomon, Int. J. Mod. Phys. C 7, 595 (1996); D. Sonette, R. Cont, J. Phys. I Fance 7, 431 (1997). 1. A. Dăgulescu, V.M. Yakovenko, cond-mat/1432, Eu. Phys. J. B 17, 723 (2). 11. The U.S. Census data, The PSID Web site, Statistics of Income 1997, Individual Income Tax Retuns, Pub. 134, Rev (IRS, Washington DC, 1999). See stats/soi/. 14. P. Saile, M. Webe, Household and Individual Income Data fom Tax Retuns (IRS, Washington DC, 1998), T. Petska, M. Studle, R. Petska, Futhe Examination of the Distibution of Individual Income and Taxes Using a Consistent and Compehensive Measue of Income (IRS, 2), W. Felle, An Intoduction to Theoy and Its Applications, Vol. 2 (John Willey, New Yok, 1966) p D.H. Weinbeg, A Bief Look at Postwa U.S. Income Inequality, P6-191 (Census Bueau, Washington, 1996), : Foms and Instuctions (IRS, Washington, 1999). 19. H.F. Lydall, Econometica 27, 11 (1959).