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# A Generalized Preferential Attachment Model for Business Firms Growth Rates: II. Mathematical Treatment

## Author

Listed:
• S. V. Buldyrev
• F. Pammolli
• M. Riccaboni
• K. Yamasaki
• D. Fu
• K. Matia
• H. E. Stanley

## Abstract

We present a preferential attachment growth model to obtain the distribution $P(K)$ of number of units $K$ in the classes which may represent business firms or other socio-economic entities. We found that $P(K)$ is described in its central part by a power law with an exponent $\phi=2+b/(1-b)$ which depends on the probability of entry of new classes, $b$. In a particular problem of city population this distribution is equivalent to the well known Zipf law. In the absence of the new classes entry, the distribution $P(K)$ is exponential. Using analytical form of $P(K)$ and assuming proportional growth for units, we derive $P(g)$, the distribution of business firm growth rates. The model predicts that $P(g)$ has a Laplacian cusp in the central part and asymptotic power-law tails with an exponent $\zeta=3$. We test the analytical expressions derived using heuristic arguments by simulations. The model might also explain the size-variance relationship of the firm growth rates.

## Suggested Citation

• S. V. Buldyrev & F. Pammolli & M. Riccaboni & K. Yamasaki & D. Fu & K. Matia & H. E. Stanley, 2006. "A Generalized Preferential Attachment Model for Business Firms Growth Rates: II. Mathematical Treatment," Papers physics/0609020, arXiv.org.
• Handle: RePEc:arx:papers:physics/0609020
as

File URL: http://arxiv.org/pdf/physics/0609020

## References listed on IDEAS

as
1. Canning, D. & Amaral, L. A. N. & Lee, Y. & Meyer, M. & Stanley, H. E., 1998. "Scaling the volatility of GDP growth rates," Economics Letters, Elsevier, vol. 60(3), pages 335-341, September.
2. De Fabritiis, G. & Pammolli, F. & Riccaboni, M., 2003. "On size and growth of business firms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 38-44.
3. Yamasaki, Kazuko & Matia, Kaushik & Buldyrev, Sergey V. & Fu, Dongfeng & Pammolli, Fabio & Riccaboni, Massimo & Stanley, H. Eugene, 2004. "Preferential attachment and growth dynamics in complex systems," MPRA Paper 15908, University Library of Munich, Germany, revised 06 Feb 2006.
4. L. A. N. Amaral & S. V. Buldyrev & S. Havlin & H. Leschhorn & P. Maass & M. A. Salinger & H. E. Stanley & M. H. R. Stanley, 1997. "Scaling behavior in economics: I. Empirical results for company growth," Papers cond-mat/9702082, arXiv.org.
5. Stephen Hymer & Peter Pashigian, 1962. "Firm Size and Rate of Growth," Journal of Political Economy, University of Chicago Press, vol. 70, pages 556-556.
6. Buldyrev, S.V & Dokholyan, N.V & Erramilli, S & Hong, M & Kim, J.Y & Malescio, G & Stanley, H.E, 2003. "Hierarchy in social organization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 330(3), pages 653-659.
7. Hall, Bronwyn H, 1987. "The Relationship between Firm Size and Firm Growth in the U.S. Manufacturing Sector," Journal of Industrial Economics, Wiley Blackwell, vol. 35(4), pages 583-606, June.
8. S. V. Buldyrev & L. A. N. Amaral & S. Havlin & H. Leschhorn & P. Maass & M. A. Salinger & H. E. Stanley & M. H. R. Stanley, 1997. "Scaling behavior in economics: II. Modeling of company growth," Papers cond-mat/9702085, arXiv.org.
Full references (including those not matched with items on IDEAS)

### JEL classification:

• L25 - Industrial Organization - - Firm Objectives, Organization, and Behavior - - - Firm Performance
• D21 - Microeconomics - - Production and Organizations - - - Firm Behavior: Theory
• O47 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - Empirical Studies of Economic Growth; Aggregate Productivity; Cross-Country Output Convergence
• D39 - Microeconomics - - Distribution - - - Other
• L00 - Industrial Organization - - General - - - General
• L60 - Industrial Organization - - Industry Studies: Manufacturing - - - General
• L65 - Industrial Organization - - Industry Studies: Manufacturing - - - Chemicals; Rubber; Drugs; Biotechnology; Plastics
• L16 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Industrial Organization and Macroeconomics; Macroeconomic Industrial Structure
• E17 - Macroeconomics and Monetary Economics - - General Aggregative Models - - - Forecasting and Simulation: Models and Applications

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