Scale-invariant properties of public-debt growth

Public debt is one of the important economic variables that quantitatively describes a nation’s economy. Because bankruptcy is a risk faced even by institutions as large as governments (e.g., Iceland), national debt should be strictly controlled with respect to national wealth. Also, the problem of eliminating extreme poverty in the world is closely connected to the study of extremely poor debtor nations. We analyze the time evolution of national public debt and find “convergence”: initially less-indebted countries increase their debt more quickly than initially more-indebted countries. We also analyze the public debt-to-GDP ratio R, a proxy for default risk, and approximate the probability density function P (R) with a Gamma distribution, which can be used to establish thresholds for sustainable debt. We also observe “convergence” in R: countries with initially small R increase their R more quickly than countries with initially large R. The scaling relationships for debt and R have practical applications, e.g. the Maastricht Treaty requires members of the European Monetary Union to maintain R< 0.6. Copyright c © EPLA, 2010 Introduction. – Just as an individual is expected to control his/her debt to asset ratio, so is a government expected to control its national debt as a function of the country’s wealth, measured, e.g., by its gross domestic product (GDP). In a dynamic global economy, excessive borrowing cannot persist indefinitely, as creditors are bound to call in large loans. While a country suffers financial problems when its GDP does not increase fast enough, even more serious trouble begins when its debt increases faster than its GDP. While national GDP has been the topic of many studies on economic growth [1–3], the empirical analysis of public debt has lagged due to lack of comprehensive data. Large sets of public-debt data, dating back several decades, and ranging from poor to rich countries, have recently become available. Here we use concepts of statistical physics to analyze public-debt data for a wide cross-section of economies including underdeveloped, developing, and developed countries. The total publicdebt data, along with total GDP data, are available at the Inter-American Development Bank [4], and are compiled and analyzed in refs. [5,6]. Population data are available (a)E-mail: [email protected] by theWorld Bank, and can be reconstructed through the GDP and per capita GDP data compiled in ref. [7]. We deflate all USD amounts to units of the USD in the year 2000. In our analysis, we compare public debt only within the same country, in order to avoid any differences in the theoretical and practical definition of debt and the reporting of debt by various countries, an issue pointed out in refs. [5,8]. Hence, our results are robust with respect to mis-reporting and ambiguous definitions of public debt [8]. In fig. 1 we plot the debt-to-GDP ratio R for many countries, grouped by common historical, geographical, and financial factors. In fig. 1(f) we plot the average R calculated for three subgroups, according to World Bank Income Group (IG) classifications, and observe relatively high levels of R among the poorest countries. Economic-growth theories predict that GDP should “converge” towards equality, with wealthy countries experiencing smaller relative growth rates than poor countries. However, the opposite has been found for economic-wealth data [9–11]. So we address the question, what are the growth dynamics for public debt? To answer this question, we analyze a comprehensive database of national public debt and GDP to investigate the dynamics of debt growth and growth in R.