Models for the size distribution of businesses in a price driven market

Abstract

Abstract. A microscopic model of aggregation and fragmentation is introduced to investigate the size distribution of businesses. In the model, businesses are constrained to comply with the market price, as expected by the customers, while customers can only buy at the prices offered by the businesses. We show numerically and analytically that the size distribution scales like a power-law. A mean-field version of our model is also introduced and we determine for which value of the parameters the mean-field model agrees with the microscopic model. We discuss to what extent our simple model and its results compare with empirical data on company sizes in the U.S. and debt sizes in Japan. Finally, possible extensions of the mean-field model are discussed, to cope with other empirical data.

Democracy versus dictatorship in self-organized models of financial markets

“…Both models exhibit stationary cluster size distributions that are similar to those obtained in the original models [10,14,15]. In this section, we assume that the cluster size distribution n s /N 0 and the character distribution Q(p) are independent.…”

“…Hence, the size distribution of the clusters follows a power law corresponding to α = 5/2, with an exponential cutoff, as in the original models [10,15]. For a = 0, the exponential cut-off vanishes.…”

“…The first term on the right-hand side describes the appearance of s clusters of size 1 by fragmentation of a cluster of size s while the second term describes the disappearance of clusters of size 1 by coagulation with another cluster. As shown in [15], the cluster size distribution is given by…”

“…To have a system in a critical state, the average size of a transaction should diverge. As shown in [15], the average size of a transaction is given by 1/a. Hence, the democratic model is self-organized but the dictatorship model is not.…”

“…The proposed models are extensions of the model of Eguíluz and Zimmermann [14,15], which is itself a dynamical formulation of a model by Cont and Bouchaud [10]. Initially, every agent is given a random number p between 0 and 1, chosen from a uniform distribution.…”