We define generalized Pareto curves as the curve of inverted Pareto coefficients b(p), where b(p) is the ratio between average income or wealth above rank p and the p-th quantile Q(p) (i.e. b(p) = E[X|X > Q(p)]/Q(p)). We use them to characterize entire distributions, including places like the top where power laws are a good description, and places further down where they are not. We develop a method to nonparametrically recover the entire distribution based on tabulated income or wealth data as is generally available from tax authorities, which produces smooth and realistic shapes of generalized Pareto curves. Using detailed tabulations from quasi-exhaustive tax data, we demonstrate the precision of our method both empirically and analytically. It gives better results than the most commonly used interpolation techniques. Finally, we use Pareto curves to identify recurring distributional patterns, and connect those findings to the existing literature that explains observed distributions by random growth models.

Jean-Jacques Rousseau wrote in the Preface to his famous Discourse on Inequality that “I consider the subject of the following discourse as one of the most interesting questions philosophy can propose, and unhappily for us, one of the most thorny that philosophers can have to solve. For how shall we know the source of inequality between men, if we do not begin by knowing mankind?” (Rousseau, 1754). This citation of Rousseau appears in an article in Spanish where Dagum (2001), in the memory of whom this book is published, also cites Socrates who said that the only useful knowledge is that which makes us better and Seneca who wrote that knowing what a straight line is, is not important if we do not know what rectitude is. These references are indeed a good illustration of Dagum’s vast knowledge, which was clearly not limited to the ?eld of Economics. For Camilo the ?rst part of Rousseau’s citation certainly justi?ed his interest in the ?eld of inequality which was at the centre of his scienti?c preoccupations. It should however be stressed that for Camilo the second part of the citation represented a “solid argument in favor of giving macroeconomic foundations to microeconomic behavior” (Dagum, 2001). More precisely, “individualism and methodological holism complete each other in contributing to the explanation of individual and social behavior” (Dagum, 2001).

What is Lorenz Curve A graphical representation of the distribution of wealth or income is known as the Lorenz curve, and it is used in the field of economics. Max O. Lorenz came up with the idea in 1905 for the purpose of providing a representation of the unequal distribution of income. How you will benefit (I) Insights, and validations about the following topics: Chapter 1: Lorenz curve Chapter 2: Gini coefficient Chapter 3: Normal distribution Chapter 4: Pareto distribution Chapter 5: Log-normal distribution Chapter 6: Jensen's inequality Chapter 7: Income inequality metrics Chapter 8: Logistic distribution Chapter 9: Coefficient of variation Chapter 10: Laplace distribution Chapter 11: Lévy distribution Chapter 12: Jaccard index Chapter 13: Theil index Chapter 14: Atkinson index Chapter 15: Mean absolute difference Chapter 16: Generalized Pareto distribution Chapter 17: Log-logistic distribution Chapter 18: Lorenz asymmetry coefficient Chapter 19: Dagum distribution Chapter 20: Noncentral beta distribution Chapter 21: Log-Cauchy distribution (II) Answering the public top questions about lorenz curve. (III) Real world examples for the usage of lorenz curve in many fields. Who this book is for Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of Lorenz Curve.

Since the publication of the first edition over 30 years ago, the literature related to Pareto distributions has flourished to encompass computer-based inference methods. Pareto Distributions, Second Edition provides broad, up-to-date coverage of the Pareto model and its extensions. This edition expands several chapters to accommodate recent result

​This book presents a rigorous treatment of the mathematical instruments available for dealing with income distributions, in particular Lorenz curves and related methods. The methods examined allow us to analyze, compare and modify such distributions from an economic and social perspective. Though balanced income distributions are key to peaceful coexistence within and between nations, it is often difficult to identify the right kind of balance needed, because there is an interesting interaction with innovation and economic growth. The issue of justice, as discussed in Thomas Piketty’s bestseller “Capital in the Twenty-First Century” or in the important book “The Price of Inequality” by Nobel laureate Joseph Stiglitz, is also touched on. Further, there is a close connection to the issue of democracy in the context of globalization. One highlight of the book is its rigorous treatment of the so-called Atkinson theorem and some extensions, which help to explain under which type of societal utility functions nations tend to operate either in the direction of more balance or less balance. Finally, there are some completely new insights into changing the balance pattern of societies and the kind of coalitions between richer and poorer parts of society to organize political support in democracies in either case. Oxford University's Sir Tony Atkinson, well known for his so-called Atkinson theorem, writes in his foreword to the book: “[The authors] contribute directly to t he recent debates that are going on in politics. [...] with this book the foundation of arguments concerning a proper balance in income distribution in the sense of identifying an ‘efficient inequality range’ has got an additional push from mathematics, which I appreciate very much.”