Clustering processes
Consider modeling a large number of economic agents or units interacting in a market. The nature of the agents or units involved varies according to the context of the model. They could be groups of households, firms or some sections or divisions of firms, sectors of an economy, or even a collection of national economies.
All are considered to be made up of some basic units (of capital stocks measured in some basic units, sales measured in millions of dollars, number of employees in thousands, and so on). These units or agents in a collection, possibly of some fixed size, can be arranged or arrange themselves in a number of ways, and clusters or groupings of basic units of many different sizes may emerge or come to exist in markets, some temporarily and some permanently.We treat such models as decomposable random combinatorial structures, as we shortly explain. Some existing results, such as the Herfindahl index in the literature on industrial organization, are rephrased in this language. The reader may wish to consult Sutton (1997), for example, for summaries of some of the related topics in the more traditional industrial organization literature.
In this book, we associate types with choices by or characteristics of agents, and speak of agents forming groups by types, by associating types with choices, or by characteristics. The word, “choice” should therefore be interpreted broadly. Choices may be decisions where to buy goods, or what to buy, by households or firms, which algorithms to use for search for employment by unemployed workers, which types of layoff policies to implement by firms, which policy stance to take by central banks, and so on.[14]
As briefly mentioned in earlier sections, we derive probability distributions of the sizes of such clusters or groups and of their growth rates, such as stationary- state or time-dependent probability distributions of firm sizes or market shares.
This can be done analytically in simple models, or by simulation otherwise. By having time-dependent distributions we can assess growth rates of developing economies and new industries at initial or later stages, steady-state distributions of market shares of mature markets, and so on.[15]To derive probability distributions for the numbers of clusters of various sizes, we count the total number of possible configurations of a set of objects of some fixed size and, usually, assign probabilities equally That is, we treat each basic configuration or pattern as equiprobable. More on this later. We obtain some nonstationary distributions by the method of probability generating functions or cumulant generating functions. In some applications, we translate behavioral models into Fokker-Planck equations via Langevin equations, as in Chapter 8.
Consider economic models in which n agents individually choose, usually subject to some possibly imperfectly perceived externalities or aggregate effects of choices made by others, one of Kn possible alternatives. The number of choices may depend on n, whence the subscript on K. Depending on the context of the model, Kn may be finite, as in binary choice models, such as in Aoki (1996a, pp. 143, 166), or countably infinite. The number of types, that is, the number of choices, may or may not be known in advance, or may be random. Models discussed here are more general than those in the traditional discretechoice literature. For one thing, we explicitly consider dynamic processes due to the nature of random choices, as will become evident as we proceed.[16]
When a collection of agents falls into distinct groups, there often is no natural way of labeling the groups. The situation is akin to those of occupancy problems in probability textbooks: A large number of indistinguishable balls are placed in a large number of unmarked identical-looking boxes. To produce sensible limiting behavior as the numbers of agents and types become very large, we resort to order statistics, that is, we label groups in order of decreasing size.
This is sensible because a rather small number of groups may dominate the aggregate behavior, such as market excess demands for some goods. Such an example will be presented in Chapter 12.We keep the total number of agents finite, at least in the initial stage of model formulation. We eventually may let the number increase, to link our results with those in the macroeconomic literature. We use the finitary approach in model building in the sense of Costantini and Garibaldi (1999). It is convenient to normalize the number of agents of each type by the number of agents of all types. Then, one speaks of the fractions of agents of various types, where all fractions sum to one. Fractions describe compositions of agents by types. This “demographic” picture of the distribution of agents by types is given by a vector of fractions, which is a point of a simplex of some dimension.[17]
Arratia and Tavare (1994) have shown that many random combinatorial objects, which are decomposable into some basic units, have a component structure whose joint distribution is equal to, or approximated by, that of a process of independent random variabes, conditional on the value of a weighted sum of the variables. In particular, they have shown that Poisson random variables work for a class of combinatorial structures called assemblies, such as permutations, random mapping functions, and partitions of sets; that negative binomial random variables work for a class of structures called multisets, such as mapping patterns and partitions of integers; and that binomial random variables work for a class called selections.
Partition vectors characterize random exchangeable partitions, and also random combinatorial structures. We use partition vectors as state vectors of these combinatorial objects. More concretely, let n denote the “size,” broadly interpreted, of a market to be modeled. The structure of the market is characterized by the partition vector
Here, ai := ai (n) is the number of components, also called clusters, of size i, so that
is the number of components or types in the market.
For example, ai may represent the number of firms with i units of sales per year in some industry, the number of sectors of an economy with i firms, or the number of countries that produce i units of some goods. We also note from definition that
is the total size of the market, to be interpreted as the total number of participants in the market, or total capitalization in the market, total sales per unit time of a sector of, economy, and so on, depending on particular contexts of models.
The vector a is an example of a partition vector, mentioned in Chapter 2, to describe the state of configurations of combinatorial structures.
Our main objective is to derive probability distributions for sizes of clusters. To treat decomposable combinatorial structures randomly, we first count the total number of possible configurations with the same partition vector. This allows us to assign probabilities to each configuration by treating each equally, that is we assign probabilities in proportion to the number of configurations. This is the equiprobable assignment of probabilities referred to earlier. We may also tilt probability distributions by weighing configurations not equally but with different weights to allow some features of structures to affect the probability assignment differently Some examples are in later sections.
10.2