Introduction

In this paper, I consider an approach to modeling certain kinds of game situations that is somewhat different from the standard noncooperativegame approach. Roughly speaking, the situations have the following features in common: (1) a large number of players; (2) repeated partitioning of the player set over time into small, randomly selected groups; (3) gamelike interaction of the members of each group over a brief span of time; and (4) extensive knowledge by each player about the past history of actions taken by aggregates from the population, but limited information about the past history of actions taken by identifiable individuals in the populations. I have already applied this approach to an election model (Rosenthal (1982)) and, jointly with Henry Landau, to two bargaining models (Rosenthal and Landau (1979, 1981)). (In addition, Shefrin (1981) has worked on a related approach, with a view toward economic applications. An early version of this paper antedated and stimulated my work in this general area.) My goals in this chapter are to describe the approach (first spelled out in Rosenthal (1979)), its applicability, its advantages and disadvantages relative to alternative approaches, and also to discuss some side issues. In keeping with the spirit of this volume, however, I concentrate on the bargaining models.

Because the actions of individuals in the situations under consideration have little influence on the population aggregates, the approach assumes implicitly that individuals neglect this influence in making their decisions.

Introduction

In this chapter we have as initial data a production set Y, a collection of preference relations ≳1,…, ≳n and a commodity vector ω ᄐ Rℓ. After defining the notion of attainable allocation, we introduce the concepts of optimality in the sense of Pareto and of price equilibrium. This chapter is devoted to the study of the interrelationships of these two concepts. Under general continuity hypotheses, Section 4.2 gathers basic definitions and the well-known facts for the convex preferences case. Under smoothness, but not convexity, assumptions, Sections 4.3 and 4.4 deal, respectively, with local first- and second-order theory, and Section 4.5 contains some results on approximate global supportability of optima by prices. Finally, Section 4.6 investigates the simplest facts about the topological and manifold structure of the set of optimal allocations.

Introduction

In this and the next chapter an extensive account is given of the Walrasian price equilibrium theory of exchange and production. In comparison to the previous chapter, the key new element is that consumers are now characterized not only by their preferences but also by their initial endowments of commodities. To streamline the exposition, the present chapter concentrates on economies without production, whereas the next goes into any distinct issue raised by the latter.

Section 5.2 presents the basic definitions of consumers, exchange economies, and exchange price equilibrium. This is done both for the general and the smooth case. Different systems of equations whose zeroes describe the equilibrium state are presented. One, the excess uitility map, is in the spirit of the previous chapter. We shall nevertheless emphasize an approach via excess demand functions. This is done partly because situations with a continuum of consumers can be accommodated at almost no cost.

Placing ourselves in a smooth framework, Section 5.3 presents the central concept of regular economy. Roughly speaking, an economy is regular if the relevant systems of equations, for example, the excess demand function, is not (first-order) degenerate, that is, singular, at equilibrium. In Section 5.8 and more deeply in chapter 8, we shall argue that, in a precise sense, nonregular economies are pathological. At any rate, it is for regular economies that the full power of the differentiate approach can be displayed. Thus, for example, the equilibria of a regular economy are well determined in the sense of being locally unique and persistent under perturbations of the economy. This is shown in Section 5.4 by an implicit function theorem argument that amounts, in essence, to the verification that the number of equations and unknowns is the same. Regular economies turn out thus to be the proper setting for the rigorous application of this classical technique.

In this chapter we introduce the basic concept of the production set of an economy and discuss some of its properties (Section 3.2). We shall describe how the production set can be viewed as the aggregate of a population of firm-specific technologies (Section 3.3). Associated with each production set we will define a number of derived, but nevertheless important, concepts such as the distance function, the normal manifold, and the profit function (Section 3.4). Finally, smoothness concepts and hypotheses (Sections 3.5 and 3.6), examples (Section 3.7), and a notion of proximity for production sets (Section 3.8) are discussed.

Production sets and efficient productions

The technological possibilities open to an economy are represented by a set Y ⊂ Rℓ, called the production set, of feasible input-output, or production, vectors. If y ᄐY, then it is understood that it is technologically possible to produce the output vector defined by max {yi, 0} by using the input vector defined by max{–yi, 0).

Until the end of World War II mathematical economics was almost synonymous with the application of differential calculus to economics. It was on the strength of this technique that the mathematical approach to economics was initiated by Cournot (1838) and that the theory of general economic equilibrium was created by Walras (1874) and Pareto (1909). Hicks's Value and Capital (1939) and Samuelson's Foundations of Economic Analysis (1947) represent the culmination of this classical era.

After World War II general equilibrium theory advanced gradually toward the center of economics, but the process was accompanied by a dramatic change of techniques: an almost complete replacement of the calculus by convexity theory and topology. In the fundamental books of the modern tradition, such as Debreu's Theory of Value (1959), Arrow and Harm's General Competitive Analysis (1971), Scarf's Computation of Equilibrium Prices (1982), and Hildenbrand's Core and Equilibria of a Large Economy (1974), derivatives either are entirely absent or play, at most, a peripheral role.

Why did this change occur? Appealing to the combined impact of Leontief's input-output analysis, Dantzig and Koopmans's linearprogramming, and von Neumann and Morgenstern's theory of games, would be correct but begs the question. Schematizing somewhat (or perhaps a great deal), we could mention two internal weaknesses of the traditional calculus approach that detracted from its rigor and, more importantly, impeded progress.

This chapter has two purposes. The first is to make good on the repeated promise to justify our attention to regular objects by establishing their typicality or, in the terminology of this chapter, their genericity. Hence, we shall argue that whenever a property has been called regular, the term is deserved. The second purpose is to illustrate by reiterated application the uses of transversality theory, an important mathematical technique that can be informally described to economists as a sophisticated version of the old counting of equations and unknowns.

A general introduction to the generic point of view is presented in Section 8.2. Section 8.3 describes the formal setting underlying the mathematical transversality theorems. Those are then applied to an investigation of the generic structure of demand (Section 8.4), production (Section 8.5), optima (Section 8.6), equilibria (Section 8.7), and some aspects of the equilibrium correspondence (Section 8.8). The aim is not to be exhaustive but to discuss a repertoire of typical situations and useful tricks. It is part of the objective of the chapter to make clear that, although they may differ in degree of complexity, most genericity arguments are, at bottom, very similar.

We define and study in this chapter the characteristics of an individual consumer. In particular, we introduce the concepts of preference relation, utility function, and demand function.

For a considerable part of this book it would suffice to accept the notion of demand function as the definitional characteristic of consumers. It should be emphasized, however, that demand functions are a poor conceptual foundation for economic theory and that the grounding of the latter on preferences goes beyond an aesthetic convenience. Demand functions are good devices for the study of price equilibrium theory but are inadequate for the analysis of welfare issues (see Chapter 4) or theoretical problems that do not emphasize prices (see, for example, Chapter 7).

Our aim in this chapter is not merely to get the existence of demand functions. We want them to be smooth. Mathematically speaking, a demand function exhibits nothing but the parametric dependence of a maximizer element.

Introduction

In this chapter we study economies with nontrivial production possibilities. The notions of equilibrium and regular equilibrium will be defined and the corresponding index theorem established.

Most of the analysis and results of the previous chapter could be generalized without much difficulty to cover the production case. It would, however, be pointless, not to say dull, to devote this chapter to doing so in detail. Hence, we shall limit ourselves to present the basic concepts and dwell only on those aspects that are specific to the presence of production.

Definitions of production economy, equilibrium, regular equilibrium, and index are presented in Sections 6.2 and 6.3. Section 6.4 is devoted to the special important case of constant returns economies. The index relation for production economies is stated and proved in Section 6.5. Section 6.6. parallels Section 5.7 and spells out the implications of production for the uniqueness of equilibrium. The point will be made that to get the latter the presence of production is helpful and definitely not a complicating factor.

This chapter gathers a number of mathematical definitions and theorems that, to different extents, will be needed for the economic theory of the following seven chapters. It is not meant to be read systematically before the rest of the book. The chapter is divided into twelve sections. The ordering of the sections is one of convenience and not of intrinsic importance for later developments. This being a book on differentiable techniques, it stands to reason that the central sections are Sections B on linear algebra, C on differential calculus, D on optimization, H on differentiable manifolds, I on transversality theory, and J on degrees of functions and indices of zeros of vector fields.

The economic theory of this book is presented in Chapters 2 through 8 with a fairly strict adherence to the axiomatic method and without presupposing much. No similar claim is made, however, for this chapter. A quick reading of the headings of the different sections will convince the reader that their content cannot be a systematic, complete, or rigorous exposition. It serves to fix terminology and to facilitate reference, but if one wishes to go deeper into the purely mathematical aspects, this chapter is no substitute for the study of the pertinent mathematical sources cited at the end of every section. It should suffice to say that the chapter does not contain any proofs.

We saw in Chapter 5 that the Walrasian allocations of an economy are optima or, in more descriptive terms, that they exhaust the gains from trade. The converse is, of course, not true. This raises the following question: Among the optima, which further properties characterize Walrasian equilibria? This chapter is devoted to investigating this problem in the context of exchange economies. Under the hypothesis of a continuum of agents we shall obtain a number of important results. In Walrasian theory individual agents optimize with respect to price vectors given independently of their own actions. Hence, the proper reference framework of the theory is one where single agents lack any macroscopic significance. The continuum hypothesis embodies this requirement, and it is thus natural that it should be an essential ingredient of a characterization of equilibria.

In Section 7.2 we shall offer precise definitions for the following three properties of an allocation x:

1. (i) No group of traders can allocate their initial endowment vector among themselves in a manner unanimously preferred (core property);

2. (ii) no individual trader would be better off with the net trade of any other trader (anonymity property); and

3. (iii) individual agents appropriate all the gains from trade they contribute (no-surplus property).

The three concepts introduced

The Wealth of Nations is said to have been the first systematic treatise upon the subject of economics, and students of that problem field should not be allowed to forget that it had its origin as a section in Adam Smith's lectures on jurisprudence. Its author referred to the subject as “the science of the legislator” (1976: 19, 428). To see the aptness of this characterization of “the science of political economy,” one need only review the circumstances of its application in the actual practice of its prominent practitioners since the days of Smith. In his most characteristic role as practitioner, the economist is a specialist advisor to legislators and citizens in a legislative frame of mind. The advising of business firms and other administrative organizations or agencies with well-defined ends to attain is an altogether different activity. In the role by which he is familiarly known in the history of the subject, he has practiced his profession as counselor to legislators in their deliberations upon how well or ill an economic system is working and upon how it might be modified to improve its performance.

It is in this factual setting, in which the economist will have assumed his characteristic role of legislative advisor, that we may identify the basic concepts–the categories of things and relations that the reasoning and talk are all about. Three of these classes of things may be noted immediately.

This book is a collection of previously published essays. It makes sense as a collection because these essays have more to say together than the sum of each individually. That collective message is largely methodological. It concerns the possibility of a style of economic theory rather different from the current stock in most economic journals.

What mainly differentiates these articles from most economic theory today is their willingness to entertain the consequences of new assumptions. Most of the best economic theory of the last forty years has concerned the consequences of application of supply and demand theory to new areas (such as resource economics, the demand and supply of patents, the economics of discrimination, and the economics of foreign exchange markets) and also to the solution of classical problems (such as proof of the existence of equilibrium and the Pareto optimality of that equilibrium in a classical Walrasian model).

In contrast, these essays represent an alternative approach to the advance of economic theory. That alternative approach is to explore the consequences of new behavioral assumptions. To give one example of this exploration, prior to the publication of “The Market for ‘Lemons’” there was very little work on the economic consequences of imperfect information (except in formal game theory). This omission is curious because information imperfections are consistent with economists' utilitarian view of the world.

Since the publication of The Wealth of Nations, economists have built an entire profession on a single powerful theory of human behavior based on a few simple assumptions. That model has been fruitfully applied to a wide range of problems.

But, while economists have been elaborating their analysis, keeping their basic behavioral assumptions the same, sociologists, anthropologists, political scientists, and psychologists have been developing and validating models based on very different assumptions.

For most types of economic behavior, the economists' model is probably quite adequate. The models developed by other social scientists are generally ill-suited for direct incorporation into economic analysis. Nevertheless, insofar as studies in these other disciplines establish that people do not behave as economists assume they do, economics should endeavor to incorporate these observations.

This paper presents an example of how this might be accomplished in one special case. Psychologists have devoted considerable attention to the theory of cognitive dissonance. This theory has been used earlier by Albert Hirschman (1965) to describe attitude changes toward modernization in the course of development. Our paper expands the economic applications of cognitive dissonance and analyzes its welfare consequences in a formal model.

An overview

The basic premises

To begin, we must translate the psychological theory into concepts amenable to incorporation into an economic model. We think the theory of cognitive dissonance can be fairly represented in economists' terms in three propositions: First, persons not only have preferences over states of the world, but also over their beliefs about the state of the world.