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Theory of Zipf's Law and Beyond
Alexander I. Saichev, Yannick Malevergne, Didier Sornette
Verlag Springer-Verlag, 2009
ISBN 9783642029462 , 171 Seiten
Format PDF, OL
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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
139
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
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