World Income Distribution and Mobility

Transcription

1 ,,,, Physcs Proceda 3 (2010) Physcs Proceda 00 (2010) 1 11 Physcs Proceda World Income Dstrbuton and Moblty Beshan Xu a, Jnzhong Guo a, Nng X b, Qnghua Chen a, Yougu Wang 1a a Department of Systems Scence, School of Management, Bejng Normal Unversty, Bejng , P.R.Chna b Busness School Unversty of Shangha for Scence and Techonology, Shangha , P.R.Chna Abstract In ths paper, we show that over the perod , the world ncome dstrbuton expressed n terms of GDP per capta nvarably scales down as an exponental law. To vsualze dynamcal characterstcs behnd ths macro-stablty, we use a clock form to present the GDP per capta and rank of the 163 countres over 38 years. Correspondngly, the average varatons over tme are quantfed by absolute and relatve moblty ndexes. Furthermore, a growth moblty ndex concernng the share of each country s proposed and ts decomposton s also presented n the clock form. c 2010 Publshed by Elsever Ltd Open access under CC BY-NC-ND lcense. Keywords: GDP per capta, Dstrbuton, Clock, Moblty, Decomposton PACS: Gh, Kd, m, Ey 1. Introducton The world ncome dstrbuton of GDP per capta has been the subject of much emprcal and theoretcal work over the last two decades [1, 2, 3, 4, 5, 6, 7, 8]. Indeed, two emprcal results concernng the ncome dstrbuton have surfaced n the recent lterature: Frst, whle convergence n terms of GDP per capta has been acheved among a restrcted set of ndustralzed countres,.e. the so-called convergence club [2], dvergence has been the rule for the GDP dstrbuton taken as a whole [8]. Second, the densty functon of the cross-country GDPs dstrbuton has moved from a unmodal shape n the 1960s to a twn-peaks shape n the 1990s [5, 6, 7]. In 2003, Gulm, Gaffeo, and Gallegat presented a thrd stylzed fact regardng the world GDPs dstrbuton from a new perspectve [9]. It was the frst attempt to examne the relaton between GDP per capta and ts rank order. They found that the world ncome dstrbuton between the 30th and the 85th percentles approxmately follows a Pareto dstrbuton, and that ths result s extremely robust as movng from 1960 to Later Rok Iwahash and Tomohro 1 Author to correspondence:ygwang@bnu.edu.cn c 2010 Publshed by Elsever Ltd do: /j.phpro Open access under CC BY-NC-ND lcense.

2 1714 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Machkta looked nto ths ssue, but obtaned dfferent result [10]. He ponted out that the real cross-country GDP per capta s sgnfcantly approxmated to a geometrc sequence, furthermore, the Log GDP aganst ts rank, nstead of Log rank, dsplays a more sgnfcant relaton: It clearly fts a lne even for the rchest and poorest country groups. The studes mentoned above manly dscuss the steady world ncome dstrbuton n the macro level, however, some mcro characterstc of ntra-dstrbuton, such as the GDP per capta or rank, changes severely wth tme behnd the macro-stablty [5, 11, 12]. Ths fact s meanngful for further understandng the formaton of the dstrbuton but has seldom been notced untl Quah s work n 1993 [5]. In ths poneerng work, Quah analyzed the dynamcs n the rch cross-secton of countres ncome by a so called fractle Markov chan, and he suggested that cross-country ncomes tended towards extremes at both hgh and low endponts. Followng the approach proposed by Quah, Rchard Paap and Herman K. van Djkb examned the ndvdual swtches of countres between the rch and poor groups [11], and argued that the man moblty s from rch to poor and the mddle group between poor and rch dsappears. In addton, Francos Bourgugnon and Chrstan Morrsson extended ths work by concludng that the moblty of ndvduals n the world dstrbuton of ncome s strongly hstory dependent [12]. Those studes on the dynamcs of world ncome all focus on the varance of specfc quantles of the world dstrbuton. However, the mcro change of every country, whch s more elementary for understandng the ncome-generatng regmes, hasn t been dscussed. Besdes, consderng the measurement for moblty, there has been a lot of related work n the feld of personal ncome [14, 15, 16, 17, 18, 19, 20]. Researchers construct dstnct ndexes to measure moblty accordng to dfferent concepts, whch could also be used n measurng the world ncome moblty. We have two mprovements n ths paper. Frst, although statstcally stylzed facts are found by prevous work, t s stll controversal as to whether there s a steady law does the world ncome dstrbuton exactly obey. As a result, we check the macro-stablty of world ncome by approachng the data of GDP per capta to an exponental law usng the data from the year 1970 to Second, we further nvestgate the mcro change of world ncome by vsualzng them n a clock form advanced by Mchael Batty [13], and calculatng the total varance by a moblty ndex proposed by Felds & Ok [14]. In addton, we put forward a new ndex for measurng the moblty of world ncome and decompose t to make clear the modes or reasons of the ncome moblty. We beleve that the results shown n ths paper s mportant and useful to further understand the development and evoluton of world ncome dstrbuton. The remander s organzed as follows: Statstcal evdence examned by cross-country panel data s presented n Secton 2; The mcro change of world ncome s exhbted n Secton 3; Secton 4 gves the calculaton and decomposton of moblty; In secton 5 we conclude. 2. Data and dstrbutons We study the world ncome dstrbuton n terms of GDP per capta, usng the dataset Penn World Table verson 6.3 of Summers, Heston and Aten. Ths Table contans a set of economc tme seres, based on natonal accounts coverng 189 countres for the perod However, for observatons are not avalable for each country over the whole perod, a restrcton of the tme horzon has been mposed n order to mnmze the trade-off between the cross-secton dmenson and the tme dmenson of the panel. In our case, the sample we analyze conssts of 163 countres over the perod The world dstrbutons of GDP per capta for 163 countres over the perod n the sngle-logarthmc coordnates are presented n Fgure 1. As can be seen, almost all the curves

3 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 1: Dstrbutons of GDP per capta for 163 countres from 1970 to 2007 n dfferent years parallel wth the dashed straght lnes, whch ndcates that the GDP per capta for the sample probably follows an exponental law: G r = ae br, (1) where G r s the ncome of the country ranked r n a lst ordered by GDP per capta, a and b are the parameters of the dstrbuton. To test our hypothess, we use the exponental law to approach those dstrbutons. We fnd that equaton 1 fts the data well, whch ndcates that the per capta GDP of 163 countres and the rank obey the exponental law for all the 38 years. The estmates of the parameters for these years are collected n Table.1. To confrm our conclusons, we use the Kolmogorov-Smrnov (KS) statstcs to examne the ftness of our results. As shown n table.1, the KS statstcs are all less than 0.03, whch ndcates that t s nsuffcent to reject the concluson that the GDP per capta of these sample countres follow the exponental law. In our fttng and estmatng, some abnormal data are deleted to elmnate the nfluence of outlers, only countres whose per capta GDP s ranked 14 to 163 are taken nto consderaton. The fact that the world ncome dstrbutons follow the exponental law ndcates the macrostablty of the world ncome n terms of GDP per capta. To further verfy ths concluson, we trace another macro feature, Gn coeffcent, that characterzng the ncome nequalty. We calculate the world Gn coeffcents for all years where each country s treated smply as an nhabtant of the world, and llustrate them usng a clock form whch was frst proposed by Mchael Batty [13]. In ths form, the changes n sze for each unt are plotted n temporal clockwse drecton wth the hghest value at the centre and the lowest on the crcumference. The clock form has an advantage on the dsplayng of the overall trend, thus, we plot the data of Gn coeffcent evenly n ths form. As shown n Fgure 2, the blue curve, denotng the tme seres of Gn coeffcent from 1970 to 2007, s smooth wth the values around 0.55,.e., the Gn coeffcent remans stable over ths perod. As a result, we can conclude that the world ncome remans stable n ts macro level from a dfferent pont of vew.

4 1716 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 2: Clock form of world Gn coeffcent from 1970 to Mcro-dynamc and Clock form The above analyss shows us the stablty of world ncome n the macro level by presentng the robust world ncome dstrbuton and Gn coeffcent over tme. However, when we consder the system from a mcro pont of vew, a sngle country s GDP per capta and correspondng rank fluctuates all the tme. Take Equatoral Gunea for example, n 1970, ts per capta GDP was dollars ranked 126 n our sample countres, whle t rose to and took the 40th place n the rank order of GDP per capta n See Chna, ts per capta GDP was only dollars ranked 162 n 1970, whle n 2007, Chna s GDP per capta amounted to dollars, wth ts rank order jumpng to 74th. These drastc fluctuatons representng the mcro-dynamcs of ncome at the naton level can t be fgured out drectly from the correspondng steady dstrbutons. Therefore, our second analyss nvolves vsualzng these mcro-dynamcs or fluctuatons and examnng trajectores usng the clock form we have mentoned n secton 2. Along the dea of Batty s rank clock, we propose GDP per capta clock and correspondng rank clock to exhbt the mcro-dynamcs of country wealth under the crcumstance. In our case, a clock s a graph where characterstcs such as GDP per capta or rank orders are plotted for each country n temporal clockwse drecton wth the hghest rank or lowest value of per capta GDP at the centre and the contrary on the crcumference. Each country s colored randomly, whch provdes dstnct vsualzaton of the dynamcs. Fgure 3 s the GDP per capta clock, the dstance from the centre denotes a country s GDP per capta n the graph. Snce we plot the data wth the lowest value of per capta GDP at the centre and the largest on the crcumference, the closer a country s to the centre, the less ncome does t have. To show the mcro-dynamcs more clearly, we plot 3 representatve countres n bold curves: The bold red curve stands for the Unted States; the green one represents Luxemburg; and the blue one- Chna. We can fnd from the graph: () Most of the curves ncludng the 3 bold ones have a dvergent tendency n temporal clockwse drecton, whch ndcates that the GDP per capta of these countres have ncreased over tme, n lne wth our common sense that

5 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 3: GDP per capta clock for 163 countres from 1970 to 2007 the economy s ncreasng n the overall trend from 1970 to 2007; () The blue curve denotng Chna les closer to the centre than those for Unted States and Luxemburg, whch means Chna has almost the lowest ncome among the three selected countres. In fact, Chna has a rapd development of economy durng the 38 years, wth ts ncome amountng to n 2007, 15 tmes as t was n However, ts GDP per capta s stll much less than the other two countres due to ts large populaton; () The countres wth largest GDP per capta are mostly concentrate n Northern Europe, such as Luxembourg, Norway, Ireland, Iceland, they all le near the crcumference n the GDP per capta clock. Smlarly, we use the rank clock to present the countres dynamcs of rank orders. As s shown n Fgure 4, we equally dvde the radus nto 200, where the centre denotes the country whose GDP per capta s ranked the frst and the crcumference stands for that of the 200th. Each country s lad n the clock accordng to ts rank of GDP per capta. Snce there are too many curves wthn the clock, we use three colored bold lnes to denote the selected countres respectvely n the same way as n Fgure 3. Wth the hghest rank at the centre and the lowest on the crcumference, we could clearly see that, n 1970, Chna s GDP per capta fell far behnd the other countres, but ts economy grew fast and had a bg rse n the rank durng the 38 years. In fact, Chna s rank was 162nd n 1970 and rose up to 74th n 2007 that s a promnent progress ndeed. Whle the red lne for the Unted States remans stable near the centre, whch ndcates that there sn t any promnent change n the hgh rank for the Unted States. The GDP per capta for Luxemburg remans top-ranked from 1970 to 2007, as a result, the green lne denotng Luxemburg turns nto a dot n the centre of rank clock. 4. Moblty and ts Decomposton In the followng secton, we quantfy the mcro changes of a system over tme. We assume there are N countres n the economy labeled by an ndex set {1, 2, 3...N}. At tme k (k {0, 1}), correspondng to the ntal and fnal ponts respectvely), the ncome of country s denoted by

6 1718 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 4: Rank clock for 163 countres from 1970 to 2007 x k and the ncome dstrbuton s a vector x k = (x k 1, xk 2,...xk N ). Thus, the matrx x = (x 0, x 1 ) becomes the man subject of our analyss. As we have mentoned above, there are varous knds of ndex of moblty [14, 15, 16, 17, 18, 19, 20]. In ths paper, we employ the ndex proposed by Felds & Ok [14], whch s formulated as the followng, M(x) = 1 N x 0 x 1, (2) N =1 where N s the number of countres n the economy, x 0 and x 1 are the ntal and fnal GDP per capta or correspondng rank of county respectvely, here the ncome moblty can be regarded as ether absolute or relatve concept correspondngly. Equaton 2 smply measures the aggregate change of all countres ncome or rank. However, we argue that countres wth equal change n GDP per capta or rank may not make the same contrbuton to the overall economy. In fact, f a country provdes more essental products or servces per person,.e. has larger value of GDP per capta, we assume t would play a more mportant role and ts fluctuaton of ncome would have greater nfluence on the overall economy. Thus, countres wth dfferent ncome should not be treated equally n our measurement for moblty. Inspred by Mchael Batty s work on the defnng of growth rate n populaton, we defne a growth moblty ndex takng the share of each country nto consderaton. Frst we defne the moblty for each country at tme t, λ (t), as G (t)/g (t 1), from whch G (t) = λ (t)g (t 1) and then defne ncome shares as g (t) = G (t)/g(t), where G (t) s the GDP per capta for country at tme t and G(t) = G (t). As a result, the moblty for countres λ(t) s: λ(t) = g (t)λ (t). (3) Snce ths growth moblty ndex concerns the share of GDP per capta n each country, t suggests that countres wth larger GDP per capta have more nfluence on the overall moblty.

7 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 5: Three knds of normalzed moblty n the clock form Table.2 shows the results calculated by the three knds of moblty ndex. To vsualze and compare them, we normalze the data as follows: Y mn{y} y = max{y} mn{y}, (4) where Y s the value of moblty before normalzaton and y s the correspondng normalzed value. Therefore, we can present the ncome, rank and growth moblty n the same clock form as shown n Fgure 5. The red dotted curve s ncome moblty, the green dashed one denotes rank moblty, and the growth moblty s the blue sold one. From ths fgure, we can see: () The three curves especally the green one denotng rank moblty fluctuates drastcally durng the 38 years, whch confrms that the countres postons change fercely even as the economy s stable at macro level; () Snce the growth moblty s smaller than the rank moblty before 1991, whle t s on the contrary from the year of 1991 to 2007, we can conclude that the fluctuaton n countres wth larger value of GDP per capta contrbuted more to the overall moblty n recent years; () There s no obvous relaton between rank moblty and the other two, but the trends of ncome moblty and growth one are smlar on the whole. Ths may result from the loss of too much nformaton n the measurement of rank moblty compared wth the other two. The moblty ndex quantfes the aggregate change n each country s ncome, but t cannot reveal what les behnd t. In order to nvestgate the modes or reasons of the world ncome moblty, we decompose the moblty ndex and examne ther trajectores usng the rank clock. In equaton 3, the growth moblty λ(t) can be splt nto those assocated wth the growth and the change n the share of ncome n countres as follows: G (t) λ(t) = g (t)λ (t) = g (t) G (t 1) = { G(t) G(t 1)} g (t) g (t) g (t 1), (5) where the frst and second terms on the second lne of equaton 5 denote the overall growth Γ(t) and shft n ncome shares ϑ(t) respectvely. A more manpulable form s based on the logarthmc

8 1720 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Fgure 6: The decomposton of growth moblty growth rate log λ (t) = log [G (t)/g (t 1)], whch can be defned as: G (t) I [λ(t)] = g (t) logλ (t) = g (t) log G (t 1) = log G(t) G(t 1) + g (t) g (t) log g (t 1). (6) The log of overall growth I [Γ(t)] and log of the shares I [ϑ(t)] are the frst and second terms on the second lne of equaton 6. These growth and share varance reflect the balance between overall change and the shft of share n rank or ncome, whch n ts smplest addtve form s expressed n the log of ncome growth rates n equaton 6. And the growth and share varance over all tme perods for each data set are plotted n Fgure 6, there we observe that the red dotted curve denotng I [Γ(t)] and the blue sold one denotng I [λ(t)] have almost the same tendency durng and the green dashed curve for I [ϑ(t)] stays constantly around 0, whch ndcates that the share varance makes very lttle contrbuton to the total change whle the growth domnate, especally n the perods from 1982 to Concluson In ths paper, we studed the statstcal features of the 163 countres ncome n terms of GDP per capta durng We found that the world ncome dstrbuton could be approxmated by the exponental law. Besdes, the world Gn coeffcent remans stable around Both results ndcate that the dstrbuton of world ncome s stable durng the 38 years. However, there are volatle and turbulent mcro-dynamcs behnd such a macro-stablty, we use the clock form to llustrate these mcro changes n GDP per capta and ts correspondng rank respectvely. Furthermore, we quantfy the aggregate change of each country s ncome by the moblty ndex proposed by Felds, and tred to defne a new ndex concernng the share of each country called growth moblty. Ths new ndex allows the countes wth larger value of GDP per capta to have more nfluence on the overall moblty, as s n our common sense. Fnally, n order to

9 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) further nvestgate the modes or reasons of the ncome moblty, we decompose the moblty ndex and examne ther trajectores usng the clock and fnd that the structural growth domnates n the aggregate change of world ncome. Acknowledgments Ths research was supported by Natonal Natural Scence Foundaton of Chna under Grant of No Reference [1] M. Abramovtz, Catchng Up, Forgng Ahead and Fallng Behnd, Journal of Economc Hstory 46, (1986) [2] W. J. Baumol, Productvty Growth, Convergence and Welfare: What the Long-Run Data Show, Amencan Economc Revew 76, (1986) [3] R. J. Barro, Economc Growth n a Cross-Secton of Countres, Quarterly Journal of Economcs106, (1991), [4] N. G. Mankw, D. Romer, D.N. Wel, A Contrbuton to the Emprcs of Eco nomc Growth, Quarterly Journal of Economcs 107, (1992) [5] D. Quah, Emprcal Cross-Secton Dynamcs n Economc Growth, European Economc Revew 37, (1993a) [6] D. Quah, Galton s Fallacy and Tests of the Convergence Hypothess, Scandnavan Jounal of Economcs 95, (1993b) [7] D. Quah, Twn Peaks: Growth and Convergence n Models of Dstrbuton Dynamcs, Economc Journal 106, (1996) [8] L. Prtchett, Dvergence, Bg Tme, Journal of Economc Perspectves 11, (1997) [9] D Gulm, E. Gaffeo, M. Gallegat, Power Law Scalng n the World Income Dstrbuton, Economcs Bulletn 15, (2003) 1-7. [10] R. Iwahash, T. Machkta, A new emprcal regularty n world ncome dstrbuton dynamcs, , Economcs Bulletn 6, (2004) 1-15 [11] R. Paap, H. K. van Djk, Dstrbuton and moblty of wealth of natons, European Economc Revew 42, (1998), [12] F. Bourgugnon, C. Morrsson, The Sze Dstrbuton of Income Among World Ctzens: ,2001, Mmeo, DELTA, Pars. [13] B. Mchael, Rank clock, Nature 444, (2006) [14] G. S. Felds, E. A. Ok, Measurng Movement of Income, Economca 66, (1999), [15] S. J. Pras, Measure Socal Moblty, Journal of the Royal Statstcal, Socety A, Part I 118, (1955) [16] A. F. Shorrocks, Income Inequalty and Income Moblty, Journal of Economc Theory 19, (1978a) [17] B. R. Schller, Relatve Earnng Moblty n the Unte States, The Amercan Economc Revew 67, (1977) [18] A. F. Shorrocks, The measurement of moblty, Econometrca 46, (1978b) [19] A. B. Atknson, The Measurement of Economc Moblty, n Socal Justce and Publc Polcy, MIT Press, Cambrdge, MA, [20] Y. Wang, N. Dng, N X, Prospects for Money Transfer Models. n H. Takayasu(Eds.), Practcal Fruts of Econophyscs: Proceedngs of the Thrd Nkke Econophyscs Symposum, Sprnger-Verlag, Tokyo, 2006.

10 1722 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Appendces Table.1: Estmates of parameters and KS statstcs for the dstrbuton of GDP per capta Year â ˆb KS statstcs

11 B. Xu et al. / Physcs Proceda 3 (2010) Beshan Xu,Jnzhong Guo,X Nng,Qnghua Chen,Yougu Wang / Physcs Proceda 00 (2010) Table.2: Moblty of world ncome for over Perod Income Rank Growth