Our objectives and approaches

This book is about modeling a large collection of interacting agents, and exam­ining aggregate (deterministic) dynamics and associated stochastic fluctations.

There are two aspects or components in carrying out these objectives: dy­namics, and random combinatorial analysis.

We regard economic processes as jump Markov processes, that is, continuous­time Markov processes with at most countable state spaces, and analyze forma­tions of subgroups or clusters of agents by types. Jump Markov processes allow us to model group sentiments or pressure, such as fashion, fads, bandwagon effects, and the like. A cluster is formed by agents who use the same choices or decisions. Agents are thus identified with the rules they use at that point in time. Agents generally change their minds - that is, types - over time. This aspect is captured by specifying a set of transition rates in the jump Markov processes. Distributions of cluster sizes matter, because a few of the larger clus­ters, if and when formed, approximately determine the market excess demands for whatever goods are in the markets. There are some new approaches to firm size distributions as well.

Dynamics are represented by the master equations (the backward Chapman- Kolmogorov equations) for the joint probability distributions of suitably defined states of the collection of agents. The solutions of the master equations give us stationary or equilibrium behavior of the model and some fluctuations about them, obtained by solving the associated Fokker-Planck equations.

To fulfill our objectives, we use concepts such as partition vectors as state vectors, which arise in examining patterns formed by partitions of agents by types; Stirling numbers of the first and second kind, which have roots in combi­natorial analysis; and distributions such as the Poisson-Dirichlet distributions and the multivariate Ewens distribution. All of these are unfamiliar to tradi­tionally trained economists and graduate students of economics. We therefore present these as well as some others, as needs arise, to advance and support our modeling tasks and views proposed in this book.

Our approach is finitary. We start with a finite number of agents, each of whom is assumed to have a choice set - a set of at most countably many decision rules or behavioral rules. We define a demographic profile of agents composed of fractions of agents of the same type, with a finite number of total agents. We may let the number of agents go to infinity later, but we do not start with fractions of uncountably many agents arranged in a unit interval, for example (a typical starting point of some models in the economic literature). Our finitary approach is more work, but we reap a greater harvest of results. We may obtain more information on the natures of fluctuations, and more insight into dynamics, which get lost in the conventional approach.

Here is our approach in a nutshell. We start with a collection of a large, but finite, number of microeconomic agents in some economic context.

Then we analyze the master equations that incorporate specified transition rates. Their size effects may be important in approximate analysis of the master equations. Stationary or nonstationary solutions of the master equations are then examined to draw their economic implications.

In models that focus on the decomposable random combinatorial aspects, distributions of a few of the largest order statistics of the cluster size distributions are examined to examine their economic consequences.

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