States: Vectors of fractions of types and partition vectors

Levels of detail in describing behavior of a large collection of agents of several (or countably many) types dictate our choice of state variables.

Models are depicted in terms of configurations, namely patterns, or states, in more technical terms, and how they or some functions of them evolve with time.

At the highest level of aggregation we have macroeconomic models in terms of macroeconomic variables. At a less aggregated level, we may have sectoral models described in terms of sectoral variables, which are less aggregated than the macroeconomic variables, but are more aggregated than microeconomic variables.^[1] Stochastic description in terms of macroeconomic variables imply deterministic laws and the fluctuations about them (van Kampen 1992, p. 57).

We seek to link models for collections of microeconomic agents, whose behavior are described or stated by microeconomic specifications, with the ag­gregate or global behavior, which corresponds to mesoscopic or macroeconomic description.

1.3.1 Vectors of fractions

In this book, we use discrete states and models with finite or at most countable state spaces. This choice of state spaces is based on the way we describe micro­economic models and the details with which we describe behavior of agents - or, more pertinently, the decisions or choices they make, or the way we aggregate or incorporate microeconomic agents into macroeconomic models.

An example may help to clarify what we have in mind. At this preliminary stage of our explanation, let us suppose that agents have binary choices, or there are two types of agents, if we associate types with choices. The binary choices may be to participate or not in some joint projects, or to buy or not to buy some commodity or stocks at this point in time, etc. The nature of choices varies from model to model and from context to context. Here, we merely illustrate abstractly the ways states may be introduced. The two choices may be labeled or represented by 1 and 0, say. Then the state of n agents could be s = (s₁, s₂,, s_n), where s_i = 1 or s_i = 0, i = 1, 2,..., n.

This vector gives us a complete picture of who has chosen what. Thus, with regard to the information on the choice patterns by n agents, we don't need, nor can we have, more detail than that provided by this state vector. This is the microeconomic state at a point in time. We may then proceed to incorporate mechanisms or interaction patterns that determine how they may revise their choices over time, by specifying reward or cost structures and particulars on externalities among agents.

In some cases, we may decide not to model the collection of agents with that much detail. For example, identities of agents who have chosen 1 may not be relevant to our objectives of constructing models. We may care merely about the fraction of agents with choice 1, for example. Then, ∑_i S/n is the information we need. Then we may proceed to specify how this demographic or fractional compositional information of agents evolves with time.

With K choices or types, where K is larger than 2, detailed information on the choice pattern is provided by the vector s, where s_i now takes on one of K possible values, and choice patterns may be represented by the vector of demographic fractions, or by a vector n = (n₁, n₂,..., n_κ), where nj is the total number of agents making the j th choice.

1.3.2 Partition vectors

This choice of state vector may look natural. There is, however, another possi­bility. To understand this, let us borrow the language of the occupancy problem in probability, and think of K unmarked or indistinguishable boxes into which agents with the same choices (identical-looking balls) are placed. Let a_i be the number of boxes with i agents in them. With n agents distributed into K boxes, we have ∑n=₁ i^aj = n, and ∑^π_j=₁ aj := K_n ≤ K. The first equation counts the number of agents, and the second the number of occupied boxes.

The vector with these as as components is a state vector for some purposes. In dealing with demographic distributions such as the number of firms in various size classes, the numbers of employees, the amount of sales per month, and so on, we are not interested in the identities of firms but in the number of firms each size class, as in the histogram representations of the numbers of firms of given characteristics or categories.

In some applications, we are faced with the problem of describing sets of partitions of agents of the type called exchangeable random partitions by Kingman (1978a,b). The notion of partition vectors, in Zabell (1992), is just the right notion for discussing models in which some types of agents play a dominant role in determining market demands. This notion is discussed in detail in Chapter 3, and applied in Chapter 11 among many places.

We have briefly mentioned two alternative choices for state vectors. One of them is in terms of fractions of agents of each type or category. Instead of this more obvious choice of state variables, Watterson (1976) has proposed another way of describing states, which is less detailed than the one above using n. A level of disaggregation, or a way of describing the delabeled composition of a population of agents, is proposed that is suitable in circumstances in which new types of agents appear continually and there is no theoretical upper limit to the number of possible types. This is the so-called sampling-of-species problem in statistics (see Zabell 1993). The state of a population is described by the (unordered) set of type frequencies, i.e., fractions or proportions of dif­ferent types, without stating which frequency belongs to which type. In the context of economic modeling, this way of description does not require model builders to know in advance how many or what types of agents are in the pop­ulation. It is merely necessary to recognize that there are k distinct types in his sample of size n, and that there are aj types with j agents or represent­atives in the sample. Compositions of samples and populations at this level are given by vectors a and b with components aj and bj, respectively, such that

in the population, where N is the number of agents in the population, and K the number of distinct types, categories, or choices in the population, both being possibly infinite.

1.4