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Theory of Zipf's Law and Beyond

von: Alexander I. Saichev, Yannick Malevergne, Didier Sornette

Springer-Verlag, 2009

ISBN: 9783642029462 , 171 Seiten

Format: PDF, OL

Kopierschutz: Wasserzeichen

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Theory of Zipf's Law and Beyond


 

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