A.2 Rodolfo Benini, 261

A.3 Max Otto Lorenz, 263

A.4 Corrado Gini, 265

A.5 Luigi Amoroso, 267

A.6 Raffaele D™Addario, 269

A.7 Robert Pierre Louis Gibrat, 271

A.8 David Gawen Champernowne, 273

Appendix B Data on Size Distributions 277

Appendix C Size Distributions 283

List of Symbols 287

References 289

Author Index 319

Subject Index 327

Preface

This is a book about money, but it will not help you very much in learning how to

make money. Rather, it will instruct you about the distribution of various kinds of

income and their related economic size distributions. Speci¬cally, we have

painstakingly traced the numerous statistical models of income distribution, from

the late nineteenth century when Vilfredo Pareto developed a bold and astonishing

model for the distribution of personal income until the latest models developed some

100 years later. Our goal was to review, compare, and somehow connect all these

models and to pinpoint the unfortunate lack of coordination among various

researchers, which has resulted in the duplication of effort and waste of talent and to

some extent has reduced the value of their contributions. We also discuss the size

distributions of loss in actuarial applications that involve a number of distributions

used for income purposes. An impatient reader may wish to consult the list of

distributions covered in this book and their basic properties presented in Appendix C.

The task of compiling this interdisciplinary book took longer and was more

arduous than originally anticipated. We have tried to describe the distributions

outlined here within the context of the personalities of their originators since in our

opinion the personality, temperament, and background of the authors cited did affect

to some extent the nature and scope of their discoveries and contributions.

We hope that our readers come to regard this book as a reliable source of

information and we gladly welcome all efforts to bring any remaining errors to our

attention.

CHRISTIAN KLEIBER

Dortmund, Germany

SAMUEL KOTZ

Washington, D.C.

ix

Acknowledgments

The authors are indebted to various researchers around the globe”too numerous to

be mentioned individually”for generously providing us with preprints, reprints, and

useful advice.

Special thanks are due to Professor Giovanni Maria Giorgi for writing four

biographies of leading contributors to the ¬eld, to Professors Camilo Dagum and

Gabriele Stoppa for reading parts of the original manuscript and offering us the most

valuable suggestions and comments, to Professor Constance van Eeden and Meike

Gebel for translations from the Dutch and Italian, respectively, and to Professor

Fiorenzo Mornati for supplying important not easily accessible information about

Vilfredo Pareto. The ¬rst author would also like to thank Professor Walter Kramer¨

for his support over (by now) many years.

All of the graphs in this book were generated using the R statistical software

package (http://www.r-project.org/), the GNU implementation of the S language.

xi

CHAPTER ONE

Introduction

Certum est quia impossibile est. TERTULLIAN, 155/160 A.D.”after 220 A.D.

This book is devoted to the parametric statistical distributions of economic size

phenomena of various types”a subject that has been explored in both statistical and

economic literature for over 100 years since the publication of V. Pareto™s famous

breakthrough volume Cours d™economie politique in 1897. To the best of our

´

knowledge, this is the ¬rst collection that systematically investigates various

parametric models”a more respectful term for distributions”dealing with income,

wealth, and related notions. Our aim is marshaling and knitting together the

immense body of information scattered in diverse sources in at least eight

languages. We present empirical studies from all continents, spanning a period of

more than 100 years.

We realize that a useful book on this subject matter should be interesting, a

task that appears to be, in T. S. Eliot™s words, “not one of the least dif¬cult.” We

have tried to avoid reducing our exposition to a box of disconnected facts

or to an information storage or retrieval system. We also tried to avoid easy

armchair research that involves computerized records and heavy reliance on the

Web.

Unfortunately, the introduction by its very nature is always somewhat fragmentary

since it surveys, in our case rather extensively, the content of the volume. After

reading this introduction, the reader could decide whether continuing further study

of the book is worthwhile for his or her purposes. It is our hope that the decision will

be positive. To provide a better panorama, we have included in the Appendix brief

biographies of the leading players.

1.1 OUR AIMS

The modeling of economic size distributions originated over 100 years ago with the

work of Vilfredo Pareto on the distribution of income. He apparently was the ¬rst to

1

2 INTRODUCTION

observe that, for many populations, a plot of the logarithm of the number of incomes

Nx above a level x against the logarithm of x yields points close to a straight line of

slope Àa for some a . 0. This suggests a distribution with a survival function

proportional to xÀa , nowadays known as the Pareto distribution.

“Economic size distributions” comprise the distributions of personal incomes of

various types, the distribution of wealth, and the distribution of ¬rm sizes. We also

include work on the distribution of actuarial losses for which similar models have

been in use at least since Scandinavian actuaries (Meidell, 1912; Hagstr“m, 1925)

observed that”initially in life insurance”the sum insured is likely to be

proportional to the incomes of the policy holders, although subsequently there

appears to have been hardly any coordination between the two areas. Since the lion™s

share of the available literature comprises work on the distribution of income, we

shall often speak of income distributions, although most results apply with minor

modi¬cations to the other size variables mentioned above.

Zipf (1949) in his monograph Human Behavior and the Principle of Least

Effort and Simon (1955) in his article “On a class of skew distribution functions”

suggest that Pareto-type distributions are appropriate to model such different

variables as city sizes, geological site sizes, the number of scienti¬c publications

by a certain author, and also the word frequencies in a given text. Since the early

1990s, there has been an explosion of work on economic size phenomena in the

physics literature, leading to an emerging new ¬eld called econophysics (e.g.,

Takayasu, 2002). In addition, computer scientists are nowadays studying ¬le size

distributions in the World Wide Web (e.g., Crovella, Taqqu, and Bestavros, 1998),

but these works are not covered in this volume. We also exclude discrete Pareto-

type distributions such as the Yule distribution that have been utilized in

connection with the size distribution of ¬rms by Simon and his co-authors (see

Ijiri and Simon, 1977).

Regarding the distribution of income, the twentieth century witnessed

unprecedented attempts by powerful nations such as Russia (in 1917) and China