Eur. Phys. J. B 57, 131-138 (2007)
https://doi.org/10.1140/epjb/e2007-00165-8

A generalized preferential attachment model for business firms growth rates

II. Mathematical treatment

¹ Department of Physics,  Yeshiva University, 500 West 185th Street,  New York, NY, 10033, USA
² Faculty of Economics, University of Florence, Milan, Italy
³ IMT Institute for Advanced Studies, via S. Micheletto 3, 55100 Lucca, Italy
⁴ Tokyo University of Information Sciences, Chiba City, 265-8501, Japan
⁵ Center for Polymer Studies and Department of Physics, Boston University, Boston, MA, 02215, USA

Corresponding author: ^a buldyrev@yu.edu

Received: 31 August 2006
Revised: 13 December 2006
Published online: 13 June 2007

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 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 power-law tails with an exponent ζ = 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.