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

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  • 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.

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Paper provided by arXiv.org in its series Papers with number physics/0609020.

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Date of creation: Sep 2006
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Handle: RePEc:arx:papers:physics/0609020

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  1. De Fabritiis, Gianni & Riccaboni, Massimo & Pammolli, Fabio, 2003. "On Size and Growth of Business Firms," MPRA Paper 15866, University Library of Munich, Germany.
  2. Stephen Hymer & Peter Pashigian, 1962. "Firm Size and Rate of Growth," Journal of Political Economy, University of Chicago Press, vol. 70, pages 556.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. Bronwyn H. Hall, 1988. "The Relationship Between Firm Size and Firm Growth in the U.S. Manufacturing Sector," NBER Working Papers 1965, National Bureau of Economic Research, Inc.
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