Growth Patterns of Two Types of Macro-Models: Limiting Behavior of One-and Two-Parameter Poisson-Dirichlet Models

This paper uses novel growth models composed of clusters of heterogeneous agents, and shows that limiting behavior of oneand two-parameter PoissonDirichlet models are qualitatively very different. As model sizes grow unboundedly, the coefficients of variations of extensive variables, such as the number of total clusters, and the numbers of clusters of specified sizes all approach zero in the one-parameter models, but not in the two-parameter models. In the calculations of the coefficients of variations Mittag-Leffler distributions arise naturally. We show that the distributions of the numbers of the clusters in the models have power-law behavior. ∗Fax number 1-310-825-9528, Tel. no. 1-310-825-2360, [email protected]. The author thanks M. Sibuya for useful discussions.