Zipf s Law and Its Correlation to the GDP of Nations

Transcription

1 Celebrating 2 Years of Student Research and Scholarship 217 Zipf s La and Its Correlation to the GDP of Nations Rachel K. Skipper (Frostburg State University) Mentor: Dr. Jonathan Rosenberg, Ruth M. Davis Professor University of Maryland, College Park Abbstract This study looks at poer las, specifically Zipf s la and Pareto distributions, previously used to describe city size distribution, income distribution ithin firms, and ord distribution ithin languages and documents among other things, and Gibrat s la describing groth rate. This study seeks to discover if Zipf s la can also be used to model the distribution of GDP s orldide using Gibrat s la as a justification. The simplest method to determine Zipf s la s applicability, and the one used in this study, as to create a log log plot, plotting rank versus size of the GDPs. Using that plot, Zipf s la as verified through to criteria. First the plot must appear linear and second it must have a slope of -1. For the purpose of this study, the data looked at as for all countries and then countries split into categories of emerging economies and advanced economies for the years 25, 26, 27, and 28. The results of this study shoed that all countries and countries ith emerging economies did not appear linear on the log log plot hile advanced economies appeared linear ith a slope roughly -.7, suggesting that GDP distribution of advanced economies instead follo a Pareto distribution. Advanced economies also shoed a significantly smaller variation in groth rates over the four years as implied by Gibrat s la. This as used as a possible explanation for the distribution discovered. Introduction Problem Statement Today, countries interact constantly ith one another sending products and services around the orld. Despite these international dealings there still remains a disconnect beteen the ealthiest and the poorest countries. Some countries seem to have entirely successful and large economies hile it seems as though other nations flounder and can barely sustain their populations. A fe countries experience extreme ealth hile others survive ith extreme poverty. Although this may not be a problem for countries ith large economies, smaller economies and limited resources pose a significant dilemma for others. The implications of the massive variance beteen economies are numerous and go beyond the national level. Haddad, et. al. (23) noted that malnutrition reduction rates ithin a country folloed closely ith both household income and national levels. Filmer and Pritchett (1999) indicated by their study that income per capita and inequality of income distribution, among other factors, explain ninety-five percent of cross-national variations in mortality. Others have found that foreign trade, investment, and debt dependency, all interconnected factors in GDP, negatively affect infant mortality rates (Shen & Williamson, ). With this in mind, the problem thus emerges to explain and possibly try to mitigate the differences beteen the very ealthy countries and the much poorer countries. Purpose of Study and Research Questions With the significant impact that a country s economy size and strength has on its population, understanding the phenomena that leads to the size dispersion becomes essential. One method used to understand interaction of variables is through mathematical modeling. Interestingly some models seem to fit for a variety of phenomena spanning diverse fields. In particular, Zipf s la, a poer la, has become one of the most striking empirical facts in the social sciences and economics (Gabaix, 1999). Zipf s la has been found to be related to other knon poer las that sho up in as varied fields as the distribution of biological genera and species by Willis, size of cities by Auerbach, distribution of income by Pareto, and ord usage frequencies by Zipf (Hill). Looking at the economic implications of previous research, the purpose of this project is to apply Zipf s la to the gross domestic product of nations in order to create a theoretical frameork for the apparently vast differences beteen the fe largest economies and the many smaller ones. From this the single research question used in this study follos: Can Zipf s la be used to model the rank and size of the gross domestic product of countries?

2 218 University of Maryland The remainder of this paper is organized as follos. Chapter II gives a background including definitions of key terms and the research plan. Chapter III shos the results of the research, particularly through the use of graphs demonstrating the data. Finally, Chapter IV gives a summary of the conclusions and suggestions for future research. Background and Research Plan Background Zipf s La. One of the most prominent regularities throughout the social science disciplines is that of Zipf s la for cities (Axtell and Florida, 26). It says that the size distribution of cities ithin most countries seemingly fits a poer la. Written mathematically, this means that the probability that the size of a city is greater than some S is proportional to 1/S: P(size>S)=α/S ξ, ith ξ 1 (Gabaix, 1999). If cities are ranked by size, creating a plot of the log of the size compared to the log of the rank, one ould get a plot closely resembling a straight line ith a slope of exactly -1 (Axtell and Florida, 26). This la does not follo for some countries ith unique social structures, such as China or the former USSR, but for other developed countries Zipf s la ell approximates city size distribution (Marsili and Zhang, 1998). Although Zipf s la is most commonly used in reference to the rank and size of cities, its applications fall under a vast range of areas. In confirming Zipf s la s appearance ithin data, there are to necessary parts. First, evidence of the existence of a poer la must be shon and second, the poer la must have an exponent of -1 (Gabaix, 1999) Zipf s la is so astounding because it seems collectively society organizes itself to follo this incredibly simple distribution la ithout the expressed desires of authorities (Marsili and Zhang, 1998). Zipf s distribution essentially describes other phenomena including that of the distribution of firm sizes (Axtell and Florida, 26). In income distribution. Pareto in the late 19th century, described personal income as folloing a poer la ith an exponent of about 1.5, although after looking at several countries in 19 Gini shoed that income distributions can be estimated by poer las but ith varying exponents (Okuyama, et. al., 1999). The Pareto distribution frequently referred to as the 8-2 la, suggests that approximately tenty percent of the population controls eighty percent of the ealth (Földvári, 29). The Pareto distribution is often used to approximate Zipf s la (Meintanis, 29). Gibrat s la. Gibrat s la says that the mean and variance of the groth rate of an item are independent of its size (Hansberg, 26). Gibrat s la has been studied in relation to areas such as financial returns, firms, and city sizes (González-Val & Sanso-Navarro, 21). Gibrat s la, or the la of proportionate effect as originally named by Robert Gibrat in 1931, is used to describe surprisingly non-random and extremely complex distributions (Eeckhout, ). Gabaix (1999) described an interaction beteen Zipf s la and Gibrat s la necessary in understanding distributions of gross domestic products used in this study. Data Collection and Research Plan Data Collection. For the purposes of this study, the data used ill be gathered from the International Monetary Fund s World Economic Outlook Databases (World Economic and Financial Surveys, 21). More specifically, the data used ill be the gross domestic product of all 183 members of the International Monetary Fund from the years 25 to 28. The gross domestic product ill be measured in terms of current prices in United States dollars. This specific data as used for four main reasons. First of all, the International Monetary Fund had the most reliable and up to date data available during the time the research as completed. The database used is created biannually, beginning in January and June and appearing in publications in the folloing April and September. Secondly, the data from the IMF from the years 25 through 28 as used for its completeness. The IMF provided data on all 183 countries counted ithin its membership for these years, ith the exception of Turkmenistan in 28 and estimates for a fe countries. In the years preceding, the gross domestic product numbers for multiple countries as incomplete and in the years after 28 much more of the data ere estimates done by the IMF. Thirdly, the richness of the WEO database provided optimal data. The database provided not only the gross domestic products of the 183 countries but also provided further information about individual countries economies, specifically if the country had an emerging or advanced economy hich proved beneficial in this study.

3 Celebrating 2 Years of Student Research and Scholarship 219 Finally, the data used as measured in terms of current prices and in United States dollars to provide a constant unit of measurement for the data. Using any variation ould provide data that ould not accurately provide groth rates and unison beteen countries necessary to properly analyze the data. Research Plan. According to Gabaix (1999), to visualize Zipf s la, one orders the items being used, in this case countries, by rank (United States having the largest GDP has a rank of 1, Japan ith the second largest GDP has a rank of 2, etc.). Next a graph is dran ith the log of the rank on one axis and the log of the GDP on the other. If Zipf s la follos, a straight line ill appear. Furthermore the line ill have a slope of -1. In Gabaix s theoretical frameork for explaining Zipf s la for cities, he uses a fixed number of items groing stochastically. Then assuming that for a particular range of sizes, the items follo Gibrat s la as defined above, he concludes that those particular items, in a steady state, ill have a distribution that can be described by Zipf s la, including the poer exponent of-1 (Gabaix, 1999). With this in mind, the research plan as to follo a set of distinct steps for three different categories of data. First, the groth rates for each country ere calculated individually. Secondly, the mean groth rate and standard deviation for the data ere calculated in order to see if Gibrat s la holds for the data. Next the data ere modeled by plotting the rank versus the GDP of the countries. This alloed for a visualization of the extreme gap and the obvious preponderance of data at the loer end of the scale. After this a second plot as created, plotting the log of the rank versus the log of each country s GDP in an attempt to visualize the linear behavior of the data consistent ith a poer la. If the data appeared linear, then the final step as to determine the poer exponent by looking at the slope of the graph. If the slope is indeed -1, then Zipf s la is confirmed for the data. After this procedure as done for all countries, the countries ere split into to categories by their economies, advanced and emerging. The previous listed steps ere repeated on each of these categories of data. Results All Economies The International Monetary Fund provided information on a total of 183 countries. For the year 28, information on Turkmenistan as not available. For each of the years, 25 through 28, the mean groth rate and the standard deviation from the mean are shon in the chart in Table 1. Table 1. Years Mean Standard Deviation 25 to to to to Next, the data from each of the four years as simply plotted on the graphs in Figure 1. Each of the GDPs from all 183 countries, ith the exception of Turkmenistan in 28, are plotted on the graphs. The clear separation among GDPs can be seen in these graphs. The number of small GDPs is obviously much more heavily eighted than the number of higher GDPs.

4 University of Maryland Figure 1. Rank versus GOP of All Countries 25 Rank versus GOP of All Countries 2C6 1 1 "- <!> "- <!> Rank Rank Rank versus GOP for All Countries 27 Rank versus GOP of All Countries 2!Dl 1 1 "- <!> "- <!> Rank Rank The next step in the research process as to graph the logs of rank versus the log of the GDPs for each of the 183 countries. The graphs created are shon in Figure 2. Clearly it can be seen immediately that these do not create a roughly linear graph, immediately exposing the absence of Zipf s la among all countries. Instead of a linear graph, the points create a distinctly concave don plot. Therefore the step of finding the slope and the poer exponent is not necessary as neither Zipf s, nor the other poer las clearly apply.

5 Celebrating 2 Years of Student Research and Scholarship 1 Figure 2. Logarithm of Rank versus Logarithm of GOP For All Countries 25 Logarithm of Rank versus Logarithm of GOP for All Countries 25 o oo 2 a a. 18 "! 1 1 Logarithm of Rank Logarithm of Rank Logarithm of Rank versus Logarithm of GOP for All Countries 27 Logarithm of Rank versus Logarithm of GOP for All Countries 28 oooo a a Logarithm of Rank Logarithm of Rank Advanced Economies According to the International Monetary Fund there are a total of thirty-three countries hose economies are seen as advanced. In continuing ith the process used for all countries, the groth rates ere calculated. Then, the average groth rate and the standard deviation from that groth rate ere also calculated. The results from this calculation are shon in Table 2. Table 2. Years Mean Standard Deviation 25 to to to to Next, as as previously done, the data from the thirty-three countries considered to have advanced economies by the International Monetary Fund ere plotted and can be seen in Figure 3. Clearly among advanced countries as ell, the graphs are much more heavily eighted in the tail end of the graph. With far feer countries in this sample, each plotted point can be seen individually far more distinctly

6 2 University of Maryland Figure 3. Rank versus GDP of Countries ith Advanced Economies X Rank versus GDP of Countries ith Advanced Economies X 111 2Jj a_ 4 "' o Rank of Advanced Economies ol o Rank of Advanced Economies X Rank versus GOP for Countries ith Advanced Economies 27 Rank versus GOP for Countries ith Advanced Economies X ) '" Rank of Advanced Countries oo Ooo oooooooo Rank of Advanced Economies The third step in the process is to plot the logs of each of the ranks versus the logs of each of the GDPs for countries ith advanced economies. The graphs are shon in Figure 4 ith several fascinating things appearing. First of all, unlike the graphs of all the countries, the graphs of advanced economies do have a relatively linear distribution. On each of the graphs, the only exceptions to this linear design on the six loest ranked countries. On each of the graphs, a line as placed approximating the linearity of the top ranked tenty-seven advanced countries.

7 Celebrating 2 Years of Student Research and Scholarship 3 Figure 4. Logarithm of Rank verus Logarithm of GOP for Countries ith Advanced Economies 25 Logarithm of Rank versus Logarithm of_size for Countries ith Emerging Econom1es 23 Xl 23 e 21 2.E 19 "- (!J 18 1: g' 21 " E t Logarithm of Rank for Advanced Economies Logarithm of Rank for Advanced Economies " 2 "' a. 19 <!) 18 "i Logarithm of Rank versus Logarithm of GDP for Countries ith Advanced Economies g Logarithm of Rank versus Logarithm of GOP for Advanced Economies 23 oo oo Logarithm of Rank for Advanced Economies Logarithm of Rank for Advanced Economies 3.5 With the linearity established for the logarithm plots of the advanced economies, the final step in the process is to find the slope of these lines and thus establishing the poer exponent. For the year 25, the exponent as determined to be , for 26 the exponent as determined to be , for 27, -.73, and for 28, Emerging Economies Finally, after folloing the research steps for all countries and countries ith advanced economies, the procedure as repeated a third time, using this time instead the remaining 15 countries ith emerging economies as established by the International Monetary Fund. Once again, the groth rates ere determined for each of the countries and then the mean and standard deviation of the groth rates over each year ere determined. These results are displayed in Table 3. Table 3. Years Mean Standard Deviation 25 to to to to

8 2 University of Maryland Next, as before the rank versus GDP for the 15 countries ith emerging economies in order to display once again the much more heavily eighted loer end. Turkmenistan s economy fell into this section and so for the year 28, only 9 countries GDPs are plotted. These plots can be seen in Figure 5. Figure 5. Rank versus GOP for Countries ith Emerging Economies X Rank versus GOP of Countries 'With Emerging Economies 2!lE 2.5 c.:' ll ll. 5 1 Rank for Emerging Economies Rank of Emerging Economies 15 X i 2.5 c "e> 1.5 ll. Rank versus GDP for Countries 'With Emerging Economies 2CXJ7 Rank versus GOP of Countries ith Emerging Economies X ) i ll. 5 1 Rank of Emerging Economies Rank of Emerging Economies 15 The next step, as as done for all countries and for countries ith advanced economies, is to plot the logs of the ranks versus the logs of the GDPs of countries ith emerging economies. These can be seen in Figure 6. As ith the log plots of all countries, there is clearly once again not a linear distribution. Once again the graphs form a concave don distribution instead.

9 Celebrating 2 Years of Student Research and Scholarship 5 Figure 6. Logarithm of Rank versus Logarithm of GOp of Countries ith Emerging Economies 25 Logarithm of Rank versus Logarithm of GOP for Countries ith Emerging Economies 2f:Q;!2 18-1! 16 a_ l.:: oo!2 18 " 1! 16.e a_ - \ 1 1 Logarithm of Rank of Emerging Economies Logarithm of Rank for Emerging Economies Logarithm of Rank versus Logarithm of GOP for Countries ith Emerging Economies 27 Logarithm of Rank versus Logarithm of GOP for Countries Tth Emerging Economies 28 - E 16 m 2 c - e 18.e a. 16 I Logarithm of Rank of Emerging Economies 1 Logarit hm of Rank for Emerging Economies Conclusions and Future Research All Economies In looking at the data shon in Figure 2, e can see that the log-log graphs for each of the four years for all economies all folloed a similar pattern. In each of the four graphs, the data formed a roughly concave don pattern. As as mentioned previously in the results section, this means that all countries do not follo a poer la and therefore the second step of determining the slope is not necessary to establish Zipf s la. Although the explanation for this is not clear, a possible preliminary finding can be seen in Table 1. The standard deviation in groth rates among the 183 countries ranges beteen nine and ten percent beteen each pair of years from 25 to 28. Over the four year span the standard deviation as as high as 37 percent. Because of time constraints, it is difficult to determine hether this deviation is due to economy size. For this reason, it is quite clear that Gibrat s la also cannot be verified. Advanced Economies For advanced economies the results ere somehat different. In graphing the log of the rank versus the log of the GDP for each of the four years, a relatively linear pattern emerged. In each of the four graphs, all except approximately the loest ranked six GDPs fell roughly on the same line. From this a poer la as established. After the line estimating the data as added, the slopes for each of the four years ere determined to be approximately beteen -.7 and Since the slope deviates significantly from -1., the conclusion based on this data is that the distribution of advanced economies does not follo Zipf s la but instead follos a Pareto distribution alloing for the variation in the poer exponent. It is unclear hy the loest ranked economies seem to fall off the line.

10 6 University of Maryland A possible explanation for the Pareto distribution can be seen in Table 2. The standard deviation of groth rates beteen each pair of year ranges beteen approximately 7.5 and 9 percent hile the standard deviation across all four years as approximately 19 percent, considerably loer than that of all economies. Time constraints on the project make it difficult to determine hether this deviation is independent of size making it unclear hether Gibrat s la as verified. Emerging Economies The log-log plots for GDPs from countries ith emerging economies shoed results very similar to those of all countries. Each of the four graphs shoed a clearly concave don pattern and therefore does not represent a poer la. Because of this, Zipf s la clearly does not apply to countries ith emerging economies, making finding the slope irrelevant. In looking at the groth rates as shon in Table 3, it can be seen that the standard deviation beteen each pair of years ranges roughly beteen 1 and 11.3 percent and nearly 37 percent across all four years. This is nearly double the standard deviation beteen groth rates of advanced economies suggesting a possible explanation for hy neither Zipf s la, nor any other poer la is demonstrated through the distribution of GDPs among emerging economies. Although the large deviation exists, because of time constraints it is difficult to determine hether this deviation is related to size. Although this deviation suggests this data does not follo Gibrat s la, it is not clear hether this is in fact the case. Future Research Several avenues exist to expand upon this research and upon the knoledge concerning poer las, and in particular Zipf s la. First, since the GDP of neither all economies nor emerging economies folloed a poer la, finding a model to describe these phenomena could prove valuable in understanding hat causes this distribution. Secondly, since no clear cause for the poer la distribution of advanced economies emerged, future research ith feer time constraints could be designed in order to determine to things, if the groth rates are independent of size as in Gibrat s la and as proposed by Gabaix (1999) and if not, hat causes these economies to follo this pattern. A theoretical design could provide ansers that this research as not able to provide. Finally, research can be done to continually expand upon the number of knon distributions for hich Zipf s la accurately models. References Axtell, R., Florida, R. (26). Emergent cities: micro-foundations of Zipf s la. Eeckhout, J. (). Gibrat s la for (all) cities. The American Economic Revie. 94 (5), Filmer, D., Pritchett, L. (1999). The impact of public spending on health: does money matter? Social Science & Medicine. 49, Földvári, P. (29). Estimating income inequality from the tax data ith a priori assumed income distributions in Hungary, Historical Methods. 42(3), Gabaix, X. (1999). Zipf s la for cities: an explanation. The Quarterly Journal of Economics. 1(3), González-Val, R., Sanso-Navarro, M. (21). Gibrat s la for countries. Journal of Population Economics, 23(4), Doi:1.17/s Haddad, L., Alderman, H., Appleton, S., Song, L., Yohannes, Y. (23). Reducing child malnutrition: ho far does income groth take us? The World Bank Economic Revie, 17(1), doi: 1.193/ber/lhg. Hill, B.M. (n.d.). A theoretical derivation of the Zipf(Pareto) la. University of Michigan. Marsili, M., Zhang, Y. (1998), Interacting individuals leading to Zipf s la. Physical Revie Letters. 8(), Meintanis, S. (29), A unified approach of testing for discrete and continuous Pareto las. Statistics Papers Statisched Hefte. 5(3), Okuyama, K., Takayasu, M., Takayasu, H., (1999). Zipf s la in income distribution of companies. Physica A Shen, C., Williamson, J.B. (). Accounting for cross-national differences in infant mortality decline ( ) among less developed countries: effects of omen s status, economic dependency, and state strength. Social Indicators Research. 53(3), doi:1.3/a: Rossi-Hansberg, E., Wright, M.L.J, (27). Urban structure and groth. Revie of Economics Studies. 74, World Economic and Financial Surveys (21). World Economic Outlook Database.