Pareto's Law for Income of Individuals and Debt of Bankrupt Companies

Abstract

We analyze the distribution of income and income tax of individuals in Japan for the fiscal year 1998. From the rank-size plots we find that the accumulated probability distribution of both data obey a power law with a Pareto exponent very close to −2. We also present an analysis of the distribution of the debts owed by bankrupt companies from 1997 to March, 2000, which is consistent with a power law behavior with a Pareto exponent equal to −1. This power law is the same as that of the income distribution of companies. Possible implications of these findings for model building are discussed.

Bankruptcy as an exit mechanism for systems with a variable number of components

“…We find that the right tail of the bad debt distribution scales as a power law with an exponent comprised between 0.7 and 1.1, and that the life-time of bankrupted firms is exponentially distributed. Our results are strikingly close to findings for Japan [14,15], suggesting that there are universal features among the underlying microscopic mechanisms responsible for exits. We further discuss a recently proposed macroeconomic model with firms' financial fragility as a main ingredient, which is able to replicate empirical observation for bankruptcies and bad debt.…”

“…The bad debt of insolvent bond issuers is distributed as a power law with α = −0.92. Our findings are strikingly close to the ones reported in [14,15] for Japanese bankrupted firms, with the bad debt for large failed firms (i.e. the right tail of the distribution) being estimated to scale down with an exponent α comprised between 0.91 and 1.…”

“…one of the causes for firms going out of business -scales as a power law. The evidence we report for bankrupted firms in Italy, Spain and France and a sample of bankrupted European bond issuers is strikingly similar to that found for Japan [14,15]. We further discuss a simple agent-based macroeconomic model based on financial fragility which, along with a power law distribution for firms size (and many other stylized facts), is capable to replicate this fact.…”

Models for the size distribution of businesses in a price driven market

“…However, taking a(s) = (ln s) β /s, we obtain that n s is a lognormal distribution for β = 1 [18]. This result is encouraging as most studies conclude that the size distribution of companies is a lognormal distribution for typical company sizes [17], and a power-law for larger sizes [19,22]. Hence, the microscopic model we introduce offers a potential mechanism to generate a size distribution with a power-law in agreement with the results of Ref.…”

“…Taking a(s) = αs −β n 1 , the asymptotic behaviour of n s is determined for every values of β. We showed that the microscopic model corresponds to β equal annual sales in the U.S. [19] and about the size of debts of bankrupt companies in Japan [22]. We also stressed that we have focussed our attention on an analytical form for a(s) inspired by the microscopic model, while other forms for a(s) can lead to a lognormal distribution for the size of the companies, an empirical result put forward by several authors [17].…”