This book is an attempt to reformulate macroeconomic modeling as currently practiced by the macroeconomic profession.

The need to improve macroeconomic models certainly is felt widely by the economic profession. A short list of the defects that we recognize in macroeconomic modeling includes extensive and almost exclusive use of the assumption of representative agents, of largely deterministic dynamic models, and inadequate attention paid to off-equilibrium dynamic phenomena. More specifically, we do not have a satisfactory model for explaining sluggish responses of macroeconomic phenomena and the distributional effects of policy actions. We lack adequate treatments of the dynamic adjustment behavior of microeconomic units in the presence of externalities of the kind designated “field effects” in this book, and are known variously in the economic literature as social influence, social consumption, or group sentiments or pressure.

This book collects my recent investigations to provide an alternative manner for building and analyzing models in macroeconomics; it is addressed to macroeconomists and advanced graduate students in macroeconomics. The book is arranged in three parts. Part I consists of three chapters. After a short introductory discussion of motivation for developing a new way to construct and analyze macroeconomic models in Chapter 1, Chapter 2 provides some simple, motivating examples of the proposed approaches. An explicitly stochastic or statistical approach is taken in Chapter 3. It collects some material that I use in the remainder of the book, since this material is not usually in the toolkit of practicing macroeconomists and is not taught in traditional economics graduate courses.

This chapter follows up the motivating examples in Sections 2.2 and 2.3, and introduces the reader to some concepts and techniques that are useful in discussing behavior of macroeconomic variables statistically, but are not currently being employed by the majority of the economics profession.

Elementary discussions of entropy, Gibbs distributions and some introductory material from large deviation theory are provided. Entropy is not only important in describing states of macroeconomic systems made up of a large collection of interacting microeconomic agents but also is crucial in providing bounds for probabilities of large deviations from normal states. Gibbs distributions are important because equilibrium distributions of macroeconomic states are of this type. Large deviation theory has obvious distributional implications in macroeconomic policy studies by means of more disaggregate models than currently practiced.

Model descriptions

Micro and macro descriptions of models

Depending on the level of detail used in explaining economic phenomena, we can have either a micro description, which is a complete description (at least theoretically) of all units or agents in the model, or a macro description, which keeps track only of some of the observables which are some functions of the microscopic or microeconomic variables in the model. Microvariables needed to achieve a complete description, in the sense that models become Markovian, usually are not observed directly (see the examples in Chapter 2).

We sometimes observe major or drastic qualitative changes in the behavioral pattern or response of economic systems. Some of these changes are apparently triggered by minor changes in economic environments or policy parameters. For convenience, we call these responses “critical phenomena,” which are economic phenomena that are overly sensitive to changes in environments or policies.

We mention two types of such phenomena. One type involves sudden changes in structure: An equilibrium may suddenly bifurcate when some model parameter crosses a critical value. Organized markets may suddenly emerge when none existed before, or may suddenly disappear. There are many examples of these types of critical phenomena.

The other type of events is also well known. We call it a piling-up phenomena. These events are often marked by crazes or fads: A significant fraction of agents suddenly have the same opinion or make a same choice, or otherwise occupy the same configuration or state, while they were more or less evenly distributed among several states in the absence of whatever triggered this sudden shift. (Recall that a state could refer to economic or expectational conditions or both.) For lack of a better term we call them piling-up of agents, or dense occupancy of a state by agents.

Sudden structural changes

There are two senses in which sudden structural changes are relevant in economic modeling. One sense is familiar. Loosely put, discontinuous or major changes are produced in some properties or characteristics of systems or models as some parameter values are changed slightly near a so-called critical value. Examples are numerous both inside and outside economics.

We collect here some items that are a further elaboration of topics discussed in the main body of this book, and to suggest potentially useful and important topics for the reader to develop further. We have consistently viewed dynamics as stochastic and we continue this view in this chapter as well.

One of the major aims of this book is to examine consequences of changing patterns of agent interaction over time. We view these patterns as distributions of agents over possible types or categories over time, and we endevour to discover stable distributions of patterns of interactions among agents.

Equilibrium distributions of patterns of classification describe stable emergent properties of the model. Some simple and suggestive examples are discussed in Chapters 4 and 5.

In models with a small number of types, choices or classes, the methods developed in Chapter 5 are quite effective. When the number of classes becomes large, (numerical) solutions of the master equations may become cumbersome. An attractive alternative is to examine equilibrium distributions directly. We outline this method in Section E.2 by drawing on the literature of population genetics and statistics.

Interactions among several types of large numbers of agents may sometimes be modeled as Markov processes on exchangeable random partitions of a set of integers. Aldous (1985) mentions J. Pitman for the name, “Chinese restaurant process” which models how a restaurant with a large number of initially empty tables is sequentially filled up as customers arrive.

In Chapter 4, we introduced jump Markov processes to pave our way for modeling a large collection of interacting agents. We concentrate here on the types of interactions or externalities that do not lend themselves to modeling with pairwise interaction. We call these externalities “field effects.” This term is chosen to convey the notion that interaction with a whole population or class of microeconomic units is involved, that is, aggregate (macroeconomic) effects are due to contributions from a whole population of microeconomic units, or, composition of the whole in the sense that fractions of units in various states or categories are involved. We defer until Chapter 6 our discussion of pairwise interaction or interactions with randomly matched (drawn) anonymous microeconomic units.

A good way to examine field effects – that is, stochastic and dynamic externality – is to study the dynamic behavior of a group of agents who interact through the choices they make in circumstances in which each agent has a set of finite decisions to choose from, and in which they may change their minds as time progresses, possibly at some cost. In such situations, an agent's choice is influenced by the vector of the fractions of agents who have selected the same decisions, because his or her perceived profit or benefit from a decision change will be a function of this vector, namely, the composition of agents with the same decisions in the whole population of agents.

There are many methods for obtaining the real number system from the rational number system. We describe one in this appendix, which “constructs” the real number system as (appropriately defined) limits of Cauchy sequences of rational numbers. An alternative constructive approach—the method of Dedekind cuts—is described in Rudin (1976). A third approach, which is axiomatic, rather than constructive, may be found in Apostol (1967), Bartle (1964), or Royden (1968).

Our presentation in this appendix, which is based on Strichartz (1982), is brief and relatively informal. For omitted proofs and greater detail than we provide here, we refer the reader to Hewitt and Stromberg (1965), or Strichartz (1982).

Construction of the Real Line

We use the following notation: ℥ will denote the set of natural numbers and ℤ the set of all integers:

ℚ will denote the set of rational numbers:

It is assumed throughout this appendix that the reader is familiar with handling rational numbers, and with the rules for addition (+) and multiplication (·) of such numbers. It can be shown that under these operations, the rational numbers form a field; that is, for all rationals a, b, and c in ℚ, the following conditions are met:

1. Addition is commutative: a + b = b + a.

2. Addition is associative: (a + b) + c = a + (b + c).

3. Multiplication is commutative: a · b – b · a.

4. Multiplication is associative: (a · b) · c = a · (b · c).

5. […]

This appendix provides a brief introduction to vector spaces, and the structures (inner product, norm, metric, and topology) that can be placed on them. It also describes an abstract context for locating the results of Chapter 1 on the topological structure on ℝ n . The very nature of the material discussed here makes it impossible to be either comprehensive or complete; rather, the aim is simply to give the reader a flavor of these topics. For more detail than we provide here, and for omitted proofs, we refer the reader to the books by Bartle (1964), Munkres (1975), or Royden (1968).

Vector Spaces

A vector space over ℝ (henceforth, simply vector space) is a set V, on which are defined two operators “addition,” which specifies for each x and y in V, an element x + y in V; and “scalar multiplication,” which specifies for each a ∈ ℝ and x ∈ V, an element ax in V. These operators are required to satisfy the following axioms for all x, y, z ∈ V and a, b ∈ ℝ:

1. Addition satisfies the commutative group axioms:

1. (a) Commutativity: x + y = y + x.

2. (b) Associativity: x + (y + z) = (x + y) + z.

3. (c) Existence of zero: There is an element 0 in V such that x + 0 = x.

4. (d) Existence of additive inverse: For every x ∈ V, there is (-x) ∈ V such that x + (-x) = 0.

This book developed out of a course I have taught since 1988 to first-year Ph.D. students at the University of Rochester on the use of optimization techniques in economic analysis. A detailed account of its contents is presented in Section 2.5 of Chapter 2. The discussion below is aimed at providing a broad overview of the book, as well as at emphasizing some of its special features.

An Overview of the Contents

The main body of this book may be divided into three parts. The first part, encompassing Chapters 3 through 8, studies optimization in n-dimensional Euclidean space, ℝ n . Several topics are covered in this span. These include—but are not limited to— (i) the Weierstrass Theorem, and the existence of solutions to optimization problems; (ii) the Theorem of Lagrange, and necessary conditions for optima in problems with equality constraints; (iii) the Theorem of Kuhn and Tucker, and necessary conditions for optima in problems with inequality constraints; (iv) the role of convexity in obtaining sufficient conditions for optima in constrained optimization problems; and (v) the extent to which convexity can be replaced with quasi-convexity, while still obtaining sufficiency of the first-order conditions for global optima.

The second part of the book, comprised of Chapters 9 and 10, looks at the issue of parametric variation in optimization problems, that is, at the manner in which solutions to optimization problems respond to changes in the values of underlying parameters.

The notion of convexity occupies a central position in the study of optimization theory. It encompasses not only the idea of convex sets, but also of concave and convex functions (see Section 7.1 for definitions). The attractiveness of convexity for optimization theory arises from the fact that when an optimization problem meets suitable convexity conditions, the same first-order conditions that we have shown in previous chapters to be necessary for local optima, also become sufficient for global optima. Indeed, even more is true. When the convexity conditions are tightened to what are called strict convexity conditions, we get the additional bonus of uniqueness of the solution.

The importance of such results, especially from a computational standpoint, is obvious. Of course, such a marked strengthening of our earlier analysis does not come free. As we show in Section 7.2, the assumption of convexity is a strong one. A function that is concave or convex must necessarily be continuous everywhere on the interior of its domain. It must also possess strong differentiability properties; for instance, all directional derivatives of such a function must exist at all points in the domain. Finally, an assumption of convexity imposes strong curvature restrictions on the underlying function, in the form of properties that must be met by its first and second-derivatives.

These results indicate that an assumption of convexity is not an innocuous one, but, viewed from the narrow standpoint of this book, the restrictive picture they paint is perhaps somewhat exaggerated.

We begin our study of optimization with the fundamental question of existence: under what conditions on the objective function f and the constraint set D are we guaranteed that solutions will always exist in optimization problems of the form max {f(x) | x ∈ D) or min {f(x) | x ∈ D}? Equivalently, under what conditions on f and D is it the case that the set of attainable values f(D) contains its supremum and/or infimum?

Trivial answers to the existence question are, of course, always available: for instance, f is guaranteed to attain a maximum and a minimum on D if D is a finite set. On the other hand, our primary purpose in studying the existence issue is from the standpoint of applications: we would like to avoid, to the maximum extent possible, the need to verify existence on a case-by-case basis. In particular, when dealing with parametric families of optimization problems, we would like to be in a position to describe restrictions on parameter values under which solutions always exist. All of this is possible only if the identified set of conditions possesses a considerable degree of generality.

The centerpiece of this chapter, the Weierstrass Theorem, describes just such a set of conditions. The statement of the theorem, and a discussion of its conditions, is the subject of Section 3.1. The use of the Weierstrass Theorem in applications is examined in Section 3.2. The chapter concludes with the proof of the Weierstrass Theorem in Section 3.3.