Buldyrev, Sergey V. and Pammolli, Fabio and Riccaboni, Massimo and Yamasaki, Kazuko and Fu, Dongfeng and Matia, Kaushik and Stanley, H. Eugene (2006): A Generalized Preferential Attachment Model for Business Firms Growth Rates: II. Mathematical Treatment. Published in: The European Physical Journal B , Vol. 57, No. 2 (May 2007): pp. 131138.

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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 socioeconomic entities. We found that P(K) is described in its central part by a power law with an exponent φ = 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 powerlaw tails with an exponent ζ = 3. We test the analytical expressions derived using heuristic arguments by simulations. The model might also explain the sizevariance relationship of the firm growth rates.
Item Type:  MPRA Paper 

Original Title:  A Generalized Preferential Attachment Model for Business Firms Growth Rates: II. Mathematical Treatment 
Language:  English 
Keywords:  firm growth, size distribution, Gibrat law, Zipf law 
Subjects:  L  Industrial Organization > L2  Firm Objectives, Organization, and Behavior > L25  Firm Performance: Size, Diversification, and Scope D  Microeconomics > D2  Production and Organizations > D21  Firm Behavior: Theory O  Economic Development, Technological Change, and Growth > O4  Economic Growth and Aggregate Productivity > O47  Measurement of Economic Growth; Aggregate Productivity; CrossCountry Output Convergence D  Microeconomics > D3  Distribution > D39  Other L  Industrial Organization > L0  General > L00  General L  Industrial Organization > L6  Industry Studies: Manufacturing > L60  General L  Industrial Organization > L6  Industry Studies: Manufacturing > L65  Chemicals; Rubber; Drugs; Biotechnology L  Industrial Organization > L1  Market Structure, Firm Strategy, and Market Performance > L16  Industrial Organization and Macroeconomics: Industrial Structure and Structural Change; Industrial Price Indices E  Macroeconomics and Monetary Economics > E1  General Aggregative Models > E17  Forecasting and Simulation: Models and Applications 
Item ID:  15980 
Depositing User:  Laknori 
Date Deposited:  01. Jul 2009 09:17 
Last Modified:  24. Feb 2013 22:47 
References:  Amaral L. A. N., Buldyrev S. V., Havlin S., Leschhorn H, Maass P., Salinger M. A., Stanley H. E. & Stanley M. H. R. (1997): “Scaling behavior in economics: I, empirical results for company growth”, Journal de Physique I France 7, 621–633. Buldyrev S. V., Amaral L. A. N., Havlin S., Leschhorn H, Maass P., Salinger M. A. , Stanley H. E. & Stanley M. H. R. (1997) “Scaling behavior in economics: II. Modeling of company growth”, Journal de Physique I France 7, 635650. Sutton J. (2002): “The variance of firm growth rates: The ‘scaling’ puzzle”, Physica A 312, 577–590. De Fabritiis G., Pammolli F., Riccaboni M., (2003): “On size and growth of business firms”, Physica A 324, 3844. Amaral L. A. N., Buldyrev S. V., Havlin S., Salinger M. A. & Stanley H. E. (1998): “Power Law Scaling for a System of Interacting Units with Complex Internal Structure”, Physical Review Letters 80, 13851388. Takayasu H. & Okuyama K. (1998): “Country dependence on company size distributions and a numerical model based on competition and cooperation”, Fractals 6, 67–79. Canning, D., Amaral, L. A. N., Lee, Y., Meyer, M. & Stanley, H. E. (1998): “Scaling the volatility of GDP growth rates”, Economics Letters 60, 335341. 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 330, 653–659. Sutton J. (1997): “Gibrat's legacy”, Journal of Economic Literature 35, 4059. Ijiri Y. & Simon H. A. (1975): “Some distributions associated with BoseEinstein statistics” , Proceedings of the National Academy of Sciences 72, 16541657. Kalecki M. (1945): “On the Gibrat distribution”, Econometrica 13, 161170. Mansfield, D. E. (1962): “Entry, innovation, and the growth of firms”, American Economic Review 52, 10241051. Hall, B. H. (1987): “The relationship between firm size and firm growth in the US manufacturing sector”, The Journal of Industrial Economics 35, 583606. Kotz, S., Kozubowski, T. J. & Podg´orski, K. (2001): The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkhauser, Boston. Reed, W. J. (2001): “The Pareto, Zipf and other power laws”, Economics Letters 74, 1519. Reed W. J., Hughes B. D. (2002): “From Gene Families and Genera to Incomes and Internet File Sizes: Why Power Laws are so Common in Nature”, Physical Review E 66, 067103. Yamasaki K., Matia K., Buldyrev S. V., Fu D., Pammolli F., Riccaboni M., and Stanley H. E. (2006): “Preferential Attachment and Growth Dynamics in Complex Systems”, Phys. Rev. E 74, 035103(R). Johnson, N. L. & Kotz, S. (1977): Urn Models and Their Applications, Wiley, New York. Kotz, S., Mahmoud, H. & Robert, P. (2000): “On generalized Polya urn models”, Statistics and Probability Letters 49, 163173. Reed, W. J. & Hughes, B. D. (2004): “A model explaining the size distribution of gene and protein families”, Mathematical biosciences 189, No. 1, 97102. Stanley, H. E. (1971): Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford. Cox, D. R. & Miller, H. D. (1968): The Theory of Stochastic Processes, Chapman and Hall, London. Gibrat R. (1930): “Une loi des reparationseconomiques: l’effet proportionnel”, Bulletin de Statistique Generale, France19, 469. Gibrat R. (1931): Les Inegalites Economiques, Librairie du Recueil Sirey, Paris. D. Fu, F. Pammolli, S. V. Buldyrev, M. Riccaboni, K. Matia, K. Yamasaki, and H. E. Stanley, (2005): “The Growth of Business Firms: Theoretical Framework and Empirical Evidence” Proc. Natl. Acad. Sci. 102, 18801 (2005) Matia, K., Amaral, L. A. N., Luwel, M., Moed, H. F. & Stanley, H. E. (2005): “Scaling phenomena in the growth dynamics of scientific output” J. Am. Soc. Inf. Sci. Technol. 56, 893902. Zipf G. (1949): Human Behavior and the Principle of Least Effort, AddisonWesley, Cambridge, MA. Hymer, S. & Pashigian, P. (1962): “Firm size and rate of growth”, Journal of Political Economy 70, 556569. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/15980 