Power Laws in Economics and Finance

A power law (PL) is the form taken by a large number of surprising empirical regularities in economics and finance. This review surveys well-documented empirical PLs regarding income and wealth, the size of cities and firms, stock market returns, trading volume, international trade, and executive pay. It reviews detail independent theoretical motivations that make sharp predictions concerning the existence and coefficients of PLs, without requiring delicate tuning of model parameters. These theoretical mechanisms include random growth, optimization, and the economics of superstars, coupled with extreme value theory. Some empirical regularities currently lack an appropriate explanation. This article highlights these open areas for future research.

[1]  Enrico Santarelli,et al.  Gibrat's Law: Are the Services Different? , 2004 .

[2]  Y. Fujiwara Zipf Law in Firms Bankruptcy , 2003, cond-mat/0310062.

[3]  George Kingsley Zipf,et al.  Human behavior and the principle of least effort , 1949 .

[4]  William A. Brock,et al.  Discrete Choice with Social Interactions , 2001 .

[5]  X. Gabaix,et al.  Why Has CEO Pay Increased so Much? , 2006 .

[6]  Costas Arkolakis Market Penetration Costs and Trade Dynamics , 2008 .

[7]  M. Marsili,et al.  Minority Games: Interacting agents in financial markets , 2014 .

[8]  Sergei Maslov,et al.  Comment on “Role of intermittency in urban development: A model of large-scale city formation” , 1998 .

[9]  J. Bouchaud,et al.  How Markets Slowly Digest Changes in Supply and Demand , 2008, 0809.0822.

[10]  Robert E. Lucas,et al.  On the Size Distribution of Business Firms , 1978 .

[11]  Sorin Solomon,et al.  Theoretical analysis and simulations of the generalized Lotka-Volterra model. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Edward E. Leamer,et al.  International Trade Theory: The Evidence , 1994 .

[13]  Moshe Levy,et al.  The Forbes 400 and the Pareto wealth distribution , 2006 .

[14]  J. Eaton,et al.  Dissecting Trade: Firms, Industries, and Export Destinations , 2004 .

[15]  Sorin Solomon,et al.  Power laws of wealth, market order volumes and market returns , 2001 .

[16]  Sergey V. Buldyrev,et al.  Scaling behavior in economics: I Epirical results for company growth , 1997, cond-mat/9702082.

[17]  Gopikrishnan,et al.  Economic fluctuations and anomalous diffusion , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  Sergey V. Buldyrev,et al.  The size variance relationship of business firm growth rates , 2008, Proceedings of the National Academy of Sciences.

[19]  Kevin J. Murphy,et al.  Compensation and Incentives: Practice vs. Theory , 1988 .

[20]  H. Stanley,et al.  The growth of business firms: theoretical framework and empirical evidence. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Yoshi Fujiwara,et al.  Do Pareto–Zipf and Gibrat laws hold true? An analysis with European firms , 2004 .

[22]  W. Vervaat On a stochastic difference equation and a representation of non–negative infinitely divisible random variables , 1979, Advances in Applied Probability.

[23]  E. Fama Mandelbrot and the Stable Paretian Hypothesis , 1963 .

[24]  H. Kesten Random difference equations and Renewal theory for products of random matrices , 1973 .

[25]  L. Benguigui,et al.  A dynamic model for city size distribution beyond Zipf 's law , 2007 .

[26]  K. Soo Zipf's law for cities: a cross-country investigation , 2005 .

[27]  Didier Sornette,et al.  Zipf's Law for Firms: Relevance of Birth and Death Processes , 2008 .

[28]  G. Yule,et al.  A Mathematical Theory of Evolution, Based on the Conclusions of Dr. J. C. Willis, F.R.S. , 1925 .

[29]  Matthew O. Jackson,et al.  Networks and Economic Behavior , 2009 .

[30]  Gilles Duranton,et al.  Some foundations for Zipf's law: Product proliferation and local spillovers , 2006 .

[31]  A. Lo,et al.  What Happened to the Quants in August 2007? , 2007 .

[32]  Andrew Cosh The Remuneration of Chief Executives in the United Kingdom , 1975 .

[33]  Steven Kou,et al.  A Diffusion Model for Growth Stocks , 2004, Math. Oper. Res..

[34]  T. Mann The Black Swan , 1954 .

[35]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[36]  V. Plerou,et al.  Scaling of the distribution of price fluctuations of individual companies. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  S. Kotz,et al.  Statistical Size Distributions in Economics and Actuarial Sciences , 2003 .

[38]  D. Zanette,et al.  ROLE OF INTERMITTENCY IN URBAN DEVELOPMENT : A MODEL OF LARGE-SCALE CITY FORMATION , 1997 .

[39]  C. I. Jones,et al.  The Shape of Production Function and the Direction of Technical Change , 2004 .

[40]  Samuel Kortum,et al.  Research, Patenting, and Technological Change , 1997 .

[41]  Mark E. J. Newman,et al.  Power-Law Distributions in Empirical Data , 2007, SIAM Rev..

[42]  M. Marsili,et al.  Interacting Individuals Leading to Zipf's Law , 1998, cond-mat/9801289.

[43]  J. Hinloopen,et al.  Comparative Advantage, the Rank-Size Rule, and Zipf's Law , 2006 .

[44]  Kevin J. Murphy,et al.  Performance Pay and Top Management Incentives , 1990 .

[45]  Joseph Persky,et al.  Retrospectives: Pareto's Law , 1992 .

[46]  R. Barro Rare Disasters and Asset Markets in the Twentieth Century , 2006 .

[47]  Benoit B. Mandelbrot,et al.  Fractals and Scaling in Finance , 1997 .

[48]  D. Sornette Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools , 2000 .

[49]  Steven N. Durlauf,et al.  Nonergodic Economic Growth , 1993 .

[50]  X. Gabaix The Granular Origins of Aggregate Fluctuations , 2009 .

[51]  Michael Biggs Strikes as Forest Fires: Chicago and Paris in the Late Nineteenth Century1 , 2005, American Journal of Sociology.

[52]  Jean-Philippe Bouchaud,et al.  Relation between bid–ask spread, impact and volatility in order-driven markets , 2006, physics/0603084.

[53]  A. Henderson‐sellers,et al.  Albedo and its importance in climate theory , 1982 .

[54]  Moshe Levy,et al.  Gibrat's Law for (All) Cities: Comment , 2009 .

[55]  Thomas Mikosch,et al.  Regularly varying functions , 2006 .

[56]  J. Chevalier,et al.  Measuring Prices and Price Competition Online: Amazon.com and BarnesandNoble.com , 2003 .

[57]  Didier Sornette,et al.  On Rational Bubbles and Fat Tails , 1999, cond-mat/9910141.

[58]  Dwight M. Jaffee,et al.  Nondiversification Traps in Catastrophe Insurance Markets , 2009 .

[59]  J. Scheinkman,et al.  Aggregate Fluctuations from Independent Sectoral Shocks: Self-Organized Criticality in a Model of Production and Inventory Dynamics , 1992 .

[60]  B. Mandelbrot STABLE PARETIAN RANDOM FUNCTIONS AND THE MULTIPLICATIVE VARIATION OF INCOME , 1961 .

[61]  H. A. Simon,et al.  Skew Distributions and the Size of Business Firms , 1977 .

[62]  Geoffrey B. West,et al.  The origin of universal scaling laws in biology , 1999 .

[63]  M. Levy,et al.  POWER LAWS ARE LOGARITHMIC BOLTZMANN LAWS , 1996, adap-org/9607001.

[64]  Sergei Maslov,et al.  Price Fluctuations from the Order Book Perspective - Empirical Facts and a Simple Model , 2001, cond-mat/0102518.

[65]  Thomas Lux,et al.  The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks , 1996 .

[66]  Glenn R. Carroll,et al.  National city-size distributions , 1982 .

[67]  B. M. Hill,et al.  A Simple General Approach to Inference About the Tail of a Distribution , 1975 .

[68]  James H. Brown,et al.  A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science.

[69]  Dennis W. Jansen,et al.  On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective , 1989 .

[70]  H Eugene Stanley,et al.  Quantifying fluctuations in market liquidity: analysis of the bid-ask spread. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[71]  Yannis M. Ioannides,et al.  Zipf’s law for cities : an empirical examination , 2000 .

[72]  H. Stanley,et al.  Scaling, Universality, and Renormalization: Three Pillars of Modern Critical Phenomena , 1999 .

[73]  Marc J. Melitz The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity , 2003 .

[74]  Robert J. Barro,et al.  Pay, Performance, and Turnover of Bank CEOs , 1990, Journal of Labor Economics.

[75]  Michael Sattinger,et al.  Assignment Models of the Distribution of Earnings , 1993 .

[76]  V. Yakovenko,et al.  Evidence for the exponential distribution of income in the USA , 2001 .

[77]  J. Eeckhout Gibrat's Law for (All) Cities , 2004 .

[78]  F. Auerbach Das Gesetz der Bevölkerungskonzentration. , 1913 .

[79]  J. Benhabib,et al.  The distribution of wealth: Intergenerational transmission and redistributive policies , 2007 .

[80]  Xavier Gabaix,et al.  The Rodney L . White Center for Financial Research A Multiplicative Model of Optimal CEO Incentives in Market Equilibrium , 2008 .

[81]  Marko Terviö Difference that CEOs Make: An Assignment Model Approach , 2007 .

[82]  Taisei Kaizoji,et al.  A precursor of market crashes: Empirical laws of Japan's internet bubble , 2005, physics/0510055.

[83]  S. Rosen The Economics of Superstars , 1981 .

[84]  Shlomo Havlin,et al.  Relation between volatility correlations in financial markets and Omori processes occurring on all scales. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[85]  PAUL EMBRECHTS,et al.  Modelling of extremal events in insurance and finance , 1994, Math. Methods Oper. Res..

[86]  LAURANCE R. DOYLE,et al.  Quantitative tools for comparing animal communication systems: information theory applied to bottlenose dolphin whistle repertoires , 1999, Animal Behaviour.

[87]  W. Reed The Pareto, Zipf and other power laws , 2001 .

[88]  W. R. Buckland,et al.  Random processes and the growth of firms , 1965 .

[89]  D. Sornette,et al.  Dynamics of book sales: endogenous versus exogenous shocks in complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[90]  S. Durlauf Complexity and Empirical Economics , 2005 .

[91]  X. Gabaix Zipf's Law for Cities: An Explanation , 1999 .

[92]  Thomas Chaney,et al.  Distorted Gravity: The Intensive and Extensive Margins of International Trade , 2008 .

[93]  Tomoya Mori,et al.  The Number Average Size Rule: A New Empirical Relationship between Industrial Location and City Size , 2008 .

[94]  C. Goldie IMPLICIT RENEWAL THEORY AND TAILS OF SOLUTIONS OF RANDOM EQUATIONS , 1991 .

[95]  Steven Brakman,et al.  An Introduction to Geographical Economics , 2001 .

[96]  John Sutton,et al.  Market Share Dynamics and the Persistence of Leadership Debate , 2004 .

[97]  Erzo G. J. Luttmer Selection, Growth, and the Size Distribution of Firms , 2007 .

[98]  M. Weitzman Subjective Expectations and Asset-Return Puzzles , 2007 .

[99]  Sherwin Rosen,et al.  Contracts and the Market for Executives , 1990 .

[100]  J. Schumpeter,et al.  Vilfredo Pareto (1848–1923) , 1949 .

[101]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[102]  Stanley,et al.  Statistical properties of share volume traded in financial markets , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[103]  J M Carlson,et al.  Highly optimized tolerance: a mechanism for power laws in designed systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[104]  Mark L. J. Wright,et al.  Establishment Size Dynamics in the Aggregate Economy , 2006 .

[105]  Jean-Philippe Bouchaud,et al.  Power laws in economics and finance: some ideas from physics , 2001 .

[106]  J. Teugels,et al.  Statistics of Extremes , 2004 .

[107]  V. Plerou,et al.  Scaling of the distribution of fluctuations of financial market indices. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[108]  R. Axtell Zipf Distribution of U.S. Firm Sizes , 2001, Science.

[109]  John L. Casti,et al.  The Theory of Networks , 1995 .

[110]  D. Champernowne A Model of Income Distribution , 1953 .

[111]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[112]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[113]  P. Krugman Confronting the Mystery of Urban Hierarchy , 1996 .

[114]  L. Amaral,et al.  Scaling behaviour in the growth of companies , 1996, Nature.

[115]  E. Helpman,et al.  Export Versus FDI with Heterogeneous Firms , 2004 .

[116]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[117]  H. Eugene Stanley,et al.  Tests of scaling and universality of the distributions of trade size and share volume: evidence from three distinct markets. , 2007 .

[118]  Peter F. Kostiuk,et al.  Firm Size and Executive Compensation , 1990 .

[119]  X. Gabaix Zipf's Law and the Growth of Cities , 1999 .

[120]  M. E. J. Newman,et al.  Power laws, Pareto distributions and Zipf's law , 2005 .

[121]  C. Mulligan Scale Economies, the Value of Time, and the Demand for Money: Longitudinal Evidence from Firms , 1997, Journal of Political Economy.

[122]  P. Whittle,et al.  A Model Explaining the Pareto Distribution of Wealth , 1957 .

[123]  Makoto Nirei,et al.  Threshold behavior and aggregate fluctuation , 2006, J. Econ. Theory.

[124]  V. Plerou,et al.  Institutional Investors and Stock Market Volatility , 2005 .

[125]  P. Cizeau,et al.  Statistical properties of the volatility of price fluctuations. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[126]  R N Mantegna,et al.  Power-law relaxation in a complex system: Omori law after a financial market crash. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[127]  Gilles Duranton,et al.  Urban Evolutions: The Fast, the Slow, and the Still , 2007 .

[128]  H. Takayasu,et al.  Zipf's law in income distribution of companies , 1999 .

[129]  Moshe Levy,et al.  Are rich people smarter? , 1997, J. Econ. Theory.

[130]  Wataru Souma,et al.  A Two Factor Model of Income Distribution Dynamics , 2007 .

[131]  B. Mandelbrot The Variation of Certain Speculative Prices , 1963 .

[132]  D. Turcotte,et al.  Forest fires: An example of self-organized critical behavior , 1998, Science.

[133]  David R. Roberts,et al.  A General Theory of Executive Compensation Based on Statistically Tested Propositions , 1956 .

[134]  Susanna C. Manrubia,et al.  Intermittency model for urban development , 1998 .

[135]  Vasco M. Carvalho Aggregate Fluctuations and the Network Structure of Intersectoral Trade , 2010 .

[136]  J. Bouchaud,et al.  Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management , 2011 .

[137]  V. Plerou,et al.  A theory of power-law distributions in financial market fluctuations , 2003, Nature.

[138]  S Solomon,et al.  Power-law distributions and Lévy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[139]  R. Mantegna,et al.  Zipf plots and the size distribution of firms , 1995 .

[140]  Moshe Levy,et al.  Market Efficiency, the Pareto Wealth Distribution, and the Lévy Distribution of Stock Returns , 2005 .

[141]  J. Córdoba,et al.  On the Distribution of City Sizes , 2003 .

[142]  Michael Mitzenmacher,et al.  A Brief History of Generative Models for Power Law and Lognormal Distributions , 2004, Internet Math..

[143]  Sergei Maslov,et al.  Dynamical optimization theory of a diversified portfolio , 1998 .