Economic Groups by the Inequality in the World GDP Distribution Ying Li Department of Management Science, School of Business, SUN YAT-SEN University, Guangzhou, 510275, China. Tel:086-20-84141020, Email: mnsliy@mail.sysu.edu.cn Hongduo Cao Department of Management Science, School of Business, SUN YAT-SEN University, Guangzhou, 510275, China.Tel:086-20-84141020, Email: caohd@mail.sysu.edu.cn Yong Tan Michael G. Foster of Business, University of Washington, Seattle, WA, 98195-3200, USA. Tel: (206) 616-6785, Fax: (206) 543-3968, Email: ytan@u.washington.edu Abstract: We find that, from 1970 to 2006, the GDPs of 181 countries are described by kinked power law distributions. The scale-free ranges, identified by distinct power law exponents, divide the countries into two economic groups of unique within-group inequality. 1992 is a time changepoint. The members in the first group before 1992 are 90~120, and 40~50 after that. Keywords: GDP; power law; scale-free; inequality; economic groups JEL classification: C40; E01; E25; E20; F01.
Economic Groups by the Inequality in the World GDP Distribution 1 Ying Li a,, Hongduo Cao a, Yong Tan b a Department of Management Science, School of Business, SUN YAT-SEN University, Guangzhou, 510275, China b Michael G. Foster of Business, University of Washington, Seattle, WA, 98195-3200, USA Abstract: We find that, from 1970 to 2006, the GDPs of 181 countries are described by kinked power law distributions. The scale-free ranges, identified by distinct power law exponents, divide the countries into two economic groups of unique within-group inequality. 1992 is a time changepoint. The members in the first group before 1992 are 90~120, and 40~50 after that. Keywords: GDP; power law; scale-free; inequality; economic groups JEL classification: C40; E01; E25; E20; F01. 1. Introduction Economic groups in the world are often distinguished by many characteristics such as economic development level, economic size, and region. Countries form these economic groups because of trade, geography, politics, history, and culture. These economic groups evolve constantly; over time, some may disappear while new ones may emerge due to new driving forces. In this paper, we analyze the GDP 2 data from 1970 to 2006 for 181 countries. Our results show that during these 37 years, although the economic development levels, development speeds, and economic policies differ across countries, and the total GDP of the world has increased, there are always two distinct scale-free ranges if the countries are ranked by their respective GDP values. Each scale-free range is characterized by a unique power law exponent, which defines the degree of inequality 3 within the range. This allows us to divide the 181 countries into two groups according to the respective scale-free ranges which they belong to. 2. Empirical findings The data in this study is retrieved from the UNCTAD Handbook of Statistics. It contains the GDP information (nominal US dollars) of 181 countries in the world for a time span from 1970 to 2006. We find that the GDP of a country and its rank follow the power law. Fig.1 plots the GDPs against their corresponding ranks on logarithmic scales. From this figure we can identify two scale-free ranges, and the division point is around the 103 th (mean) country before 1992, and 1 This work was supported in part by the National Natural Science Foundation of China (Grant No. 70801066, 71071167,71071168). Corresponding author. Tel.: +086 20 84141020 E-mail address: mnsliy@mail.sysu.edu.cn 2 We choose to analyze GDP as it represents the overall economic strength of a country. GDP per capita has been studied earlier by Di Guilmi et al (2003). We find that GDP has some quite different characteristics compared to GDP per capita. For example, all GDPs can be fitted into two piecewise power laws, while only the range between 30 th and 85 th percentiles in the distribution of GDP per capital follows power law. GDP also has larger power law exponents. 3 Power law indicates the existence of inequality whose degree is measured by the exponent.
45 th (mean) country after 1992, respectively. Before 1992, the countries from the first to around the 103 th lie well on one straight line, while the remaining countries fit another straight line, shown in Fig.1(a). After 1992, the countries from the first to around the 45 th lie almost perfectly on one straight line, while the remaining countries fit another straight line, shown in Fig.1(b). Fig. 2 shows the least squares fitted lines after dividing the countries into two scale-free ranges. Fig. 2 depicts an example of several years, but the patterns are qualitatively similar for all 37 years. Before 1992, the R 2 values of power law fitting range from 97.40% to 98.22%. After 1992, the R 2 values of power law fitting range from 98.12% to 99.14%, shown in Fig.3. (a) Before 1992 (b) After 1992 Fig. 1 Scale free ranges in a log-log plot of GDP against rank before and after 1992 (a) Before 1992 (b). After 1992 Fig. 2 The actual data and least square fitted lines four years (c ). The green solid lines plot the actual data for 1970 to 1991 while the magenta pointed lines are from the least squares fitting. The blue solid lines plot the actual data for 1992 to 2006 while the red dashed lines are from the least squares fitting. Fig.2 The actual data and least square fitted lines several years.
Fig.3 Values of R 2 for 1970 to 2006 Mathematically, the relationship between the GDP G of a country and its rank n can be expressed as: G~n where is the exponent. This is known as the power law, which corresponds to a straight line when G is plotted against n on logarithmic scales, with a slope of. The power law distribution, also known as the Pareto distribution or Zipf law, was first discovered on the distribution of incomes by Pareto in 1897. Power law distributions of macroscopic observables are ubiquitous in both natural and social sciences. Similar power laws have also been observed in many other economic phenomena, such as GDP per capita (Di Guilmi et al, 2003), Forbes 400 list of the richest people (Levy and Solomon, 1997; Klass et al, 2006), and the number of duration of recessions (Ormerod and Mounfield, 2001) In Fig. 1, we use AB to denote a typical fitted line for a given year in the first scale-free range, and BC the second. AB and BC are the corresponding slopes. Up to 1991, the estimated value of AB varies from,1.6220 to 1.9374, and its standard error falls in a range between 0.0314, and 0.0539. BC fluctuates between 6.6614 and 10.0900, and its standard error between 0.1617 and 0.2428. The average values (among 1970~1991) of AB and BC are 1.7298 and 8.0914 respectively. After 1992, the estimated value of AB varies from 1.2301 to 1.3156, and its standard error falls in a range between 0.0162 and 0.0281. BC fluctuates between 3.2406 and 3.5453, and its standard error between 0.0485 and 0.0735. The average values (among 1992~2006) of AB and BC are 1.2707 and 3.3139 respectively. All data are in Tab.1.We find that for any given year, BC is statistically significantly 4 larger than AB. Fig.4 shows AB, BC and their standard error. Tab.1 AB, BC, and turning point 1970~1991 1992~2006 AB value Standard error mean AB value Standard error mean [1.6220, 1.9374] [0.0314,0.0539] 1.7298 [1.2301, 1.3156] [0.0162,0.0281] 1.2707 BC [6.6614, 10.0900] [0.1617,0.2428] 8.0914 BC [3.2406, 3.5453] [0.0485,0.0735] 3.3139 Turning point value mean value mean 90~120 103.2727 40~50 44.6667 4 The p-value is virtually 0%.
(a). AB (b). BC Fig.4 Time path of AB, BC and their standard error in 1970~2006 The different values of divide the countries into two groups. Group 1 consists of about 45(103) countries in the first scale-free range as represented by Line AB, Group 2 includes about 136(81) countries represented by Line BC. The GDPs of the countries in Group 1 account for larger than 95 percent of the total GDP in the world. It is worth noting that this approach of grouping is a result of the two empirically observed scale-free ranges. It is not based on the economic scale itself, but rather the inequality in the world GDP distribution. For the two corresponding groups of countries, the rank of an individual country may change from year to year. 3. Discussion 1992 is an important time changepoint. Before 1992, the turning point between two groups is among 90 th to 120 th, and the average is about 103. After 1992, the turning point is among 40 th to 50 th, and the average is about 45.There is an obvious gap between two periods. The reason for that can not been decided easily. But we notice an economic crises sweeping across the world occurred in 1990 to 1992. Fig.5 Turning point
Furthermore, AB and BC jumped a lot through 1992. As the Group 1 contributes to the majority of the world GDP, we focus on the characteristics of this group, namely AB. In the following, we discuss its invariance, volatility, and differences from BC. Also, from R 2 and Tab.1 we can see the fitting after 1992 is better than that before 1992, so we mainly focus on the case after 1992. 3.1. Invariance After 1992, AB fluctuates around its average value of 1.2707 in a small range (0.0855). This implies the distribution of GDPs in Group 1 remains constant over the years. This is reflected in the fitted lines appearing nearly in parallel. This also demonstrates that the world s leading wealth increment allocation mechanism is stable, regardless of whether the global economy is in a boom or a recession, in a slack period or in a period of recovery. Although the total GDP has increased greatly in this period, the distribution roughly follows the same pattern. The relative invariance of AB illustrates the internal stability of this group. It would be interesting to investigate what causes such a phenomenon. 3.2. Volatility 5 A smaller (larger), represented as a flatter (steeper) fitted line, indicates a lower (higher) degree of distribution inequality and a smaller (larger) dispersion among the levels of economic development in different countries. AB got smaller after 1992. Therefore, the economic development gap among different countries reduced after 1992. We observe it in detail. As visible from Fig. 6, from 1992 to 1996, the economic development gap among different countries reduces; in 1996 to 2003, this gap increases; and from 2003 to 2006, the overall gap decreases again. Fig.6 Time path of AB after 1992 5 While AB varies in a small range of 0.0855, this value is larger than the standard error of AB (approximately 0.0219). Therefore, the time trend is not due to random errors. The same is true for BC.
3.3. Differences between AB and BC There is significant difference between two economic groups. The different characteristics between Lines AB and BC signify the gap between the two groups, as shown in Fig. 7. The slop represents inequality and uneven distribution of wealth. If the observed slopes of AB and BC were equal, the attenuation or development rates of the respective countries in Groups 1 and 2 should be the same. However, our analyses show that BC is statistically significantly larger than AB. This implies that the poorer countries in the world will get even poorer in a faster way compared to the richer ones. Hence, the inequality of economic development between the two groups becomes more severe. It is relatively more stable after 1992 than before 1992.We observe the case after 1992 in detail. The gap is relatively stable from 1993 to 2003, and drops during 1992 to 1993. From 2003 to 2006, the gap widens again but does not reach the level before 1992. 4. Conclusions Fig.5 Difference between slopes of the first two groups The 181 countries in the world are divided into two groups, with about 45(103) countries in the first, and the rest in the second group. This division does not consider political, historical, cultural, or geopolitical factors. Rather, it is derived from the two scale-free ranges after ranking the countries by their respective GDPs and is solely based on the characteristics of the distribution of GDPs. Among different periods, before and especially after 1992, the characteristics of the two groups are relatively stable, and the slopes of distribution functions in a log-log plot remain static over time. While inequality exists in both two groups as indicated by power law distribution, the countries in the second often have a more serious inequality problem. The world economy as a whole seems to remain stable before and after 1992, respectively, despite the constant evolution of the structure of world economy, adjustment to international economic pattern, reorganization of economic order, transformation of global industrial structure, change of international division of labor, adaptation of roles played by countries in the global
economy, and large variations between the speeds of development of different countries. This is reflected by the fact that the two relative stable scale-free ranges in the past 37 years from 1970 to 2006. 1992 is a time change point in response to the world economic crisis event. The inequality in the power to create GDP in Group 1 is quite stable. The slopes of the fitted lines in log-log plot, which depicts the GDP distribution of Group 1, fluctuate and change directions. References Di Guilmi, C., Gaffeo, E., Gallegati, M., 2003. Power Law Scaling in the World Income Distribution. Economics Bulletin 15, 1-7. Eeckhout, J., 2004. Gibrat s Law for (All) Cities. American Economic Review 94(5), 1429-1451. Eeckhout, J., 2009. Gibrat s Law for (All) Cities: Reply. American Economic Review 99(4), 1676 1683. Klass, O., Biham, O., Levy, M., Malcai, O., Solomon, S., 2006. The Forbes 400 and the Pareto Wealth Distribution. Economics Letters 90, 290-295. Levy, M., 2009. Gibrat s Law for (All) Cities: A Comment. American Economic Review 99(4), 1672 1675. Levy, M., Solomon, S., 1997. New Evidence for the Power-Law Distribution of Wealth. Physica A 242, 90-94. Ormerod, P., Mounfield, C., 2001. Power law distribution of duration and magnitude of recessions in capitalist economies: break-down of scaling. Physica A 293, 573-582. Pareto, V., 1897. Cours d Economique Politique, vol. 2.