Method of analysis

We have so far discussed how prices and the rate of interest are determined in a particular period, the present week. We have, at the same time, also discussed the movement of prices but it was only the movement of groping prices which appears in the process of tatonnement – not the movement of temporary equilibrium prices through weeks. Effective prices, at which trade is carried out, are realized at the end of each week. In order to explain the movement of effective prices from week to week, we must not confine our investigation to the present week but extend it to future weeks. This chapter is devoted to this problem which has been left unexamined in the previous chapter, that is, to the analysis of the fluctuation of prices over weeks. This is, needless to say, a very important problem which is closely related to the problem of trade cycles or economic growth.

The problem of the fluctuations in prices over weeks consists of two subproblems. The first is the comparison of different prices of a good, in a particular week, in different circumstances. The second is the intertemporal relationship between prices in different weeks. The former is usually dealt with by comparative statics. We want to discuss the latter by using the analysis of comparative dynamics.

Let us first explain how comparative statics and comparative dynamics can be applied to the analysis of the fluctuation in prices. Let us use subscript i to signify the variables and the functions in week i.

Expectations and planning

Households and firms decide their behavioural plan depending on events which are occurring in the current period and on expectations of events which will happen in the future. Their planning is not confined to the present only; they will decide on plans for the coming several weeks simultaneously with that for the current week. In the present market, however, only that part of this long-run planning which concerns current needs is carried out. It is, of course, impossible that the remaining part, concerned with the future, is carried out in the present week; that part of the planning concerned with the next week, week 1, will become effective in the next week but it will not necessarily be carried out in the same way as was decided in the present week, week 0.

Obviously one week has elapsed between week 0 when the long-run planning was decided and week 1 when the relevant part of that planning is carried out and therefore some data will have changed. Unexpected changes may occur in the individual's tastes or in the available techniques of production; also the view of future economic events may have changed during that lapse of time. Therefore as time goes by each individual and each firm will not necessarily implement the plan as it was decided. It will be examined and revised at the beginning of each week. Thus economic plans depend on expectations about the future as well as on current events.

The basic idea

Although a dynamic analysis of the behaviour of the firm has carefully been made by Hicks, 1946, it is only concerned with its production plan. In the actual world, the production plan is only a part of the whole plan which the firm makes. It includes in addition to the production plan, the demand–supply plan of the factors of production and the products, and the inventory or stock plan of these commodities. Besides, the firm will make a demand–supply plan concerning money and securities, that is, its financial plan. The purpose of this appendix is to analyse the plans systematically by a single principle of the behaviour of the firm. Especially, the problem of the demand for money is one of the central points of interest in the following analysis.

Subjective equilibrium conditions of the firm

We use the following notation. Let yi0 be the supply of product i of the firm in the present week 0, i = 2,3,…, m; yij the expected supply of the same product in week i in the future, yj0 the demand for material, capital good or factor, j, used for production in week 0, yji the expected demand for the same good or factor in week i, j = m + 1,…, n. (Throughout the following we refer to materials, capital goods, and factors of production simply and categorically as the factors of production.) These supplies and demands may differ from outputs and inputs of these commodities actually produced or carried out in the respective weeks.

The macroeconomic profession is well aware of the shortcomings of the current state of macroeconomic modeling, and is generally dissatisfied with the models it uses, as documented, for example, in Kirman (1992b) or Leijonhufvud (1993, 1995). The need to improve macroeconomic models is certainly felt widely. To cite a few examples, we do not have a satisfactory explanation of why some macroeconomic variables move sluggishly, or how policy actions affect segments of the economy differently, and we lack adequate tools for treating dynamic adjustment behavior of microeconomic units in the presence of externalities of the kind termed “social influence” by Becker (1990) and others, or in models with multiple equilibria.

Leijohnhufvud (1995) argues that the Rationality Postulate has such a strong hold on economists that, although they are aware of the notion of the Invisible Hand, a self-regulating order emerging in a complex system from the interactions of many microeconomic agents without this being part of their intentions, they are generally blind to hierarchical complexity and some of its consequences and averse to learning from other fields in which emergent properties and distributive information processing are also studied.

This book is concerned with modeling a large collection of not necessarily homogeneous microeconomic agents or units in a stochastic and dynamic framework.

This and the next chapter describe applications of a class of stochastic processes, called jump Markov processes, to model evolutionary dynamics of a large collection of interacting microeconomic units that possibly use state-dependent discrete adjustment rules. This class includes branching processes or birth-and-death processes, to which we return in Chapter 5, to model the time evolution of rival technology adoption by firms in an industry.

Market participation and other discrete adjustment behavior

The economic literature has many examples showing that optimal adjustment behavior by microeconomic units is not always continuous or small. Rather, adjustments are made at some discrete points in time by some finite magnitudes. Similar adjustment behaviors are known to be optimal in several classes of problems in finance, operations research, and control. Many of these reported results show that adjustment or decision (control) rules are of the threshold type: A specific action or decision is chosen or triggered when some key variable, which measures the gap or discrepancy between a desired or ideal value and the actual one, reaches or exceeds (becomes larger or smaller than) some preset threshold or trigger level. Adjustments can be in one direction only, as in the well-known (S, s) inventory holding rule, or they can be more generally bidirectional; that is, adjustments could be upward or downward, as for example in Caballero (1992) where firms hire or fire as the gap between actual and desired levels of employment reach lower or upper bounds.

This chapter introduces concepts and techniques that are further developed in the main body of this book, mostly via simple examples. The objective is to introduce to the reader our approaches, viewpoints, and techniques of modeling that are possibly unfamiliar to the economics profession, and to illustrate them in simple context so that basic ideas can be easily grasped. For simplification, some examples are artificial, but suggestive enough to demonstrate relevance of the notions or methods in more realistic macroeconomic models.

The examples focus on the statistical, dynamic, and state-space properties of economic variables and models. They help to introduce to the reader the notions of distinguishable and exchangeable microeconomic agents, multiplicity of microeconomic states and entropy as a measure of multiplicity, empirical distributions and related topics such as Sanov's theorem and the Gibbs conditional principle, and dynamics on trees, among others.

All of these examples are intended to suggest why certain concepts or modeling techniques, possibly not in the mainstream of the current macroeconomic literature, are useful in, or provide a new way of examining, aggregate behavior of a large number of interacting microeconomic agents.

Stochastic descriptions of economic variables

We treat all micro- and macroeconomic variables as random variables or stochastic processes, although macroeconomic variables become deterministic in the limit of the number of microeconomic units approaching infinity. Relationships among economic variables are statistical in an essential way, and are not made so by having additive disturbances or measurement errors superimposed on deterministic relationships.

This chapter discusses dynamics on state spaces that are hierarchically structured. As the examples in Sections 2.5.3 and 5.12 suggest, dynamics with such structured state spaces may exhibit sluggish responses, much slower than the exponential decays associated with ordinary dynamics.

In this chapter, we introduce the notion of the “renormalization group” into hierarchically structured state spaces as a method of aggregation, that is, as a procedure for trimming the level of branches of tree hierarchies. This aggregation scheme is different from the one we introduced via the master equation in Chapter 5, and potentially useful, but not standard in the macroeconomic literature.

We also address the question of sensitivity of macromodels with respect to details of microeconomic unit specification. By means of several examples, we show that the sluggishness of dynamic responses of hierarchically structured systems is robust, or insensitive to details of micromodel specifications.

Examples of hierarchically structured state spaces

Some examples of hierarchically structured models are presented in Chaper 2. Here are some more examples to serve as additional evidence of the relevance of hierarchically structured models in macroeconomic modeling.

Further discussion of the notion of ultrametrics, and dynamics with transition rates (probabilities) that are functions of ultrametrics follow this section.

So far, we have concentrated on modeling a large collection of microeconomic agents subject to field-effect externality. We have shown that we can specify the transition rates for microeconomic units in such a way that they satisfy the detailed balance conditions, and that we establish Gibbs distributions as equilibrium distributions for the state of finite-state Markov chains, and we have investigated the implications of multiple equilibria for the resulting macroeconomic dynamics.

In this chapter, we discuss another circumstance under which Gibbs distributions arise by modeling pairwise externality or interaction among microeconomic units. Unlike field effects, unit i is assumed to interact with unit j, which can be specified as one of / ‘s neighbors or can be anonymous, as in Cornell and Roll (1981). When more than a single pair is involved in interaction, then we have multiple-pair interactions. A history of past interaction patterns may affect the current interaction coefficients, causing complex patterns of interactions to evolve over time.

Pairwise or multiple-pair interactions

Ising model

Ising used interaction among magnetic dipoles located at neighboring sites to model phase transitions and spontaneous magnetization in 1925. See Ellis (1985, p. 131). He put magnetic dipoles at d-dimensional regular lattice sites. Later this regular lattice structure was relaxed and a graph structure was introduced with dipoles located at nodes of the graph. This type of neighborhood interaction was later formalized as a Markov random field which has a well-developed literature. Markov random field models have many applications outside physics, including sociology and economics. See Spitzer (1975) and Kindermann and Snell (1980).