Thermodynamic Limits of Macroeconomic or Financial Models: One- and Two-Parameter Poisson-Dirichlet Models

This paper examines asymptotic behavior of two types of economic or financial models with many interacting heterogeneous agents. They are oneparameter Poisson-Dirichlet models, also called Ewens models, and its extension to two-parameter Poisson-Dirichlet models. The total number of clusters, and the components of partition vectors (the number of clusters of specified sizes), both suitably normalized by some powers of model sizes, of these classes of models are shown to be related to the Mittag-Leffler distributions. Their behavior as the model sizes tend to infinity (thermodynamic limits) are qualitatively very different. In the one-parameter models, the number of clusters, and components of partition vectors are both self-averaging, that is, their coefficients of variations tend to zero as the model sizes become very large, while in the two-parameter models they are not self-averaging, that is, their coefficients of variations do not tend to zero as model sizes becomes large.