Theory of Zipf's Law and Beyond (eBook)

Alexander I. Saichev, Yannick Malevergne, Didier Sornette (Autoren)

eBook Download: PDF

2009 | 2010
XII, 171 Seiten
Springer Berlin (Verlag)
978-3-642-02946-2 (ISBN)

Infos

Zipf's law is one of the few quantitative reproducible regularities found in e- nomics. It states that, for most countries, the size distributions of cities and of rms (with additional examples found in many other scienti c elds) are power laws with a speci c exponent: the number of cities and rms with a size greater thanS is inversely proportional toS. Most explanations start with Gibrat's law of proportional growth but need to incorporate additional constraints and ingredients introducing deviations from it. Here, we present a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. First, we show that combining Gibrat's law at all rm levels with random processes of rm's births and deaths yield Zipf's law under a 'balance' condition between a rm's growth and death rate. We nd that Gibrat's law of proportionate growth does not need to be strictly satis ed. As long as the volatility of rms' sizes increase asy- totically proportionally to the size of the rm and that the instantaneous growth rate increases not faster than the volatility, the distribution of rm sizes follows Zipf's law. This suggests that the occurrence of very large rms in the distri- tion of rm sizes described by Zipf's law is more a consequence of random growth than systematic returns: in particular, for large rms, volatility must dominate over the instantaneous growth rate.

Inhalt

Preface 5
Contents 6
Symbols 9
1 Introduction 11
2 Continuous Gibrat's Law and Gabaix's Derivationof Zipf's Law 18
2.1 Definition of Continuous Gibrat's Law 18
2.2 Geometric Brownian Motion 20
2.3 Self-Similar Properties of the Geometric Brownian Motion 21
2.4 Time Reversible Geometric Brownian Motion 21
2.5 Balance Condition 22
2.6 Log-Normal Distribution 23
2.7 Gabaix's Steady-State Distribution 25
3 Flow of Firm Creation 28
3.1 Empirical Evidence and Previous Workson the Arrival of New Firms 28
3.2 Mathematical Formulation of the Flow of Firm'sBirths at Random Instants 30
3.3 Existence of a Steady-State Distribution of Firm's Sizes 33
3.4 Steady-State Density of Firm's Asset Values Obeying Gibrat's Law 35
3.5 Mean Density of Firms Younger than Age t 37
3.6 Heuristic derivation of the origin of the power law distribution of firm sizes given by Gibrat's rule 38
4 Useful Properties of Realizations of the Geometric Brownian Motion 50
4.1 Relationship Between the Distributions of Firm's Mean Ages and Sizes 50
4.2 Mean Growth vs. Stochastic Decay 52
4.3 Geometrically Transparent Definitions of Stochastically Decaying and Growing Processes 54
4.4 Majorant Curves of Stochastically Decaying Geometric Brownian Motion 56
4.5 Maximal Value of Stochastically Decaying Geometric Brownian Motion 57
4.6 Extremal Properties of Realizations of Stochastically Growing Geometric Brownian Motion 59
4.7 Quantile Curves 61
4.8 Geometric Explanation of the Steady-State Density of a Firm's Asset Value 64
5 Exit or ``Death'' of Firms 67
5.1 Empirical Evidence and Previous Workson the Exit of Firms 67
5.2 Life-Span Above a Given Level 69
5.3 Distribution of Firm's Life Durations Above a Survival Level 70
5.4 Killing of Firms upon First Reaching a Given Asset Level from Above 71
5.5 Life-Span of Finitely Living Firms 74
5.6 Influence of Firm's Death on the Balance Condition 75
5.7 Firm's Death Does Not Destroy Zipf's Law 76
5.8 Robustness Vis-a-vis the Randomness of Initial Firm's Sizes 78
6 Deviations from Gibrat's Lawand Implications for Generalized Zipf's Laws 81
6.1 Generalized Brownian Motions 82
6.1.1 Statistical Properties of Generalized GBM 82
6.1.2 Deterministic Skeleton of the Mean Density g(s) Given by a Generalized-GBM 85
6.1.3 Size Dependent Drift and Volatility 86
6.2 Diffusion Process with Constant Volatility 87
6.3 Steady-State Density of Firm's Asset Values in the Presence of Deviations from Gibrat's Law 90
6.4 Integrated Flow 92
6.5 The Semi-Geometric Brownian Motion 94
6.6 Zipf's Laws When Gibrat's Law Does Not Hold 98
7 Firm's Sudden Deaths 104
7.1 Definition of the Survival Function 104
7.2 Exponential Distribution of Sudden Deaths 105
7.3 Implications of the Existence of Sudden Firm 106
7.4 Zipf's Law in the Presence of Sudden Deaths 108
7.5 Explanation of the Generalized Balance Condition 110
7.6 Some Consequences of the Generalized Balance Condition 113
7.7 Zipf's Law as a Universal Law with a Large Basin of Attraction 114
7.8 Rate of Sudden Death Depending on Firm's Asset Value 115
7.9 Rate of Sudden Death Depending on Firm's Age 118
8 Non-stationary Mean Birth Rate 130
8.1 Exponential Growth of Firm's Birth Rate 130
8.2 Deterministic Skeleton of Zipf's Law 131
8.3 Simple Model of Birth Rate Coupled with the Overall Firm's Value 132
8.4 Generalization When Both the Initial Firm's Sizes and the Minimum Firm's Size Grow at Constant Rates 136
8.4.1 Formulation of the Model 136
8.4.2 Pdf f(s t,) of Firm's Size
8.4.3 Mean Density g(s,t) of Firm Sizes 140
8.4.4 Local Principle 142
8.4.5 Power Law Exponent and Balance Condition 143
8.4.6 Finite Lifetime of the Economy and Transitionto the Power Law Regime 144
8.5 Time-Dependence of the Average Size of the Global Economy of Firms 148
9 Properties of the Realization Dependent Distribution of Firm Sizes 153
9.1 Derivation of the Poissonian Distribution of the Number of Firms 153
9.2 Finite-Size and Statistical Fluctuation Effectson the Empirical Measurement of Zipf's Law 157
9.3 Estimation of the Distribution of Firm Sizes 158
9.4 Statistical Fluctuations of the Size of the Global Economy Using Characteristic Functions 160
10 Future Directions and Conclusions 164
10.1 Mergers and Acquisitions and Spin-offs 164
10.1.1 General Formalism 164
10.1.2 Mergers and Acquisitions and Spin-offs with Brownian Internal Growth 166
10.1.3 Mergers and Acquisitions and Spin-offs with GBM for the Internal Growth Process 168
10.2 Summary of Main Results 169
10.2.1 Importance of Balance Conditions for Zipf's Law 169
10.2.2 Essential Differences with 's Derivation of Zipf's Law 170
10.2.3 Robustness of Zipf's Law as an Attractor for Large Variance of the GMB of Firm's Growth 171
References 172
Index 176

Produkt-Details

• Erscheint lt. Verlag: 4.11.2009
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Reihe/Serie: Lecture Notes in Economics and Mathematical Systems
• Zusatzinfo: XII, 171 p. 44 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Technik
• Themenwelt: Wirtschaft ► Betriebswirtschaft / Management
• Themenwelt: Wirtschaft ► Volkswirtschaftslehre
• Schlagworte: Birth and death processes • Brownian motion • Geometric Brownian motion • Gibrat law • preferential attachment • Proportional growth • Zipf's law
• ISBN-10: 3-642-02946-9 / 3642029469
• ISBN-13: 978-3-642-02946-2 / 9783642029462