Ebook: Theory of Zipf's Law and Beyond

Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with sizes greater than S is inversely proportional to S. Zipf's law also holds in many other scientific fields. Most explanations start with Gibrat's law of proportional growth (also known as "preferential attachment'' in the application to network growth) but need to incorporate additional constraints and ingredients introducing deviations from it. This book presents a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. The general theory is presented in the language of firm dynamics for the sake of convenience but applies to many other systems. It takes into account (i) time-varying firm creation, (ii) firm's exit resulting from both a lack of sufficient capital and sudden external shocks, (iii) the coupling between firm's birth rate and the growth of the value of the population of firms. The robustness of Zipf's law is understood from the approximate validity of a general balance condition. A classification of the mechanisms responsible for deviations from Zipf's law is also offered.

---

Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with sizes greater than S is inversely proportional to S. Zipf's law also holds in many other scientific fields. Most explanations start with Gibrat's law of proportional growth (also known as "preferential attachment'' in the application to network growth) but need to incorporate additional constraints and ingredients introducing deviations from it. This book presents a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. The general theory is presented in the language of firm dynamics for the sake of convenience but applies to many other systems. It takes into account (i) time-varying firm creation, (ii) firm's exit resulting from both a lack of sufficient capital and sudden external shocks, (iii) the coupling between firm's birth rate and the growth of the value of the population of firms. The robustness of Zipf's law is understood from the approximate validity of a general balance condition. A classification of the mechanisms responsible for deviations from Zipf's law is also offered.

---

Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with sizes greater than S is inversely proportional to S. Zipf's law also holds in many other scientific fields. Most explanations start with Gibrat's law of proportional growth (also known as "preferential attachment'' in the application to network growth) but need to incorporate additional constraints and ingredients introducing deviations from it. This book presents a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. The general theory is presented in the language of firm dynamics for the sake of convenience but applies to many other systems. It takes into account (i) time-varying firm creation, (ii) firm's exit resulting from both a lack of sufficient capital and sudden external shocks, (iii) the coupling between firm's birth rate and the growth of the value of the population of firms. The robustness of Zipf's law is understood from the approximate validity of a general balance condition. A classification of the mechanisms responsible for deviations from Zipf's law is also offered.

Content:
Front Matter....Pages 1-9
Introduction....Pages 1-7
Continuous Gibrat’s Law and Gabaix’s Derivation of Zipf’s Law....Pages 9-18
Flow of Firm Creation....Pages 19-40
Useful Properties of Realizations of the Geometric Brownian Motion....Pages 41-57
Exit or “Death” of Firms....Pages 59-72
Deviations from Gibrat’s Law and Implications for Generalized Zipf’s Laws....Pages 73-95
Firm’s Sudden Deaths....Pages 97-122
Non-stationary Mean Birth Rate....Pages 123-145
Properties of the Realization Dependent Distribution of Firm Sizes....Pages 147-157
Future Directions and Conclusions....Pages 159-166
Back Matter....Pages 1-5

---

Zipf's law is one of the few quantitative reproducible regularities found in economics. It states that, for most countries, the size distributions of city sizes and of firms are power laws with a specific exponent: the number of cities and of firms with sizes greater than S is inversely proportional to S. Zipf's law also holds in many other scientific fields. Most explanations start with Gibrat's law of proportional growth (also known as "preferential attachment'' in the application to network growth) but need to incorporate additional constraints and ingredients introducing deviations from it. This book presents a general theoretical derivation of Zipf's law, providing a synthesis and extension of previous approaches. The general theory is presented in the language of firm dynamics for the sake of convenience but applies to many other systems. It takes into account (i) time-varying firm creation, (ii) firm's exit resulting from both a lack of sufficient capital and sudden external shocks, (iii) the coupling between firm's birth rate and the growth of the value of the population of firms. The robustness of Zipf's law is understood from the approximate validity of a general balance condition. A classification of the mechanisms responsible for deviations from Zipf's law is also offered.

Content:
Front Matter....Pages 1-9
Introduction....Pages 1-7
Continuous Gibrat’s Law and Gabaix’s Derivation of Zipf’s Law....Pages 9-18
Flow of Firm Creation....Pages 19-40
Useful Properties of Realizations of the Geometric Brownian Motion....Pages 41-57
Exit or “Death” of Firms....Pages 59-72
Deviations from Gibrat’s Law and Implications for Generalized Zipf’s Laws....Pages 73-95
Firm’s Sudden Deaths....Pages 97-122
Non-stationary Mean Birth Rate....Pages 123-145
Properties of the Realization Dependent Distribution of Firm Sizes....Pages 147-157
Future Directions and Conclusions....Pages 159-166
Back Matter....Pages 1-5
....