In this paper we shall study the concept of cardinal utility in three different situations (stochastic objects of choice, stochastic act of choice; independent factors of the action set) by means of the same mathematical result that gives a topological characterization of three families of parallel straight lines in a plane. This result, proved first by G. Thomsen [24] under differentiability assumptions, and later by W. Blaschke [2] in its present general form (see also W. Blaschke and G. Bol [3]), can be briefly described as follows. Consider the topological image G of a two-dimensional convex set and three families of curves in that set such that (a) exactly one curve of each family goes through a point of G, and (b) two curves of different families have at most one common point. Is there a topological transformation carrying these three families of curves into three families of parallel straight lines? If the answer is affirmative, the hexagonal configuration of Figure l(a) is observed. Let P be an arbitrary point of G, draw through it a curve of each family, and take an arbitrary point A on one of these curves; by drawing through A the curves of the other two families, we may obtain B and B’and from them C and C’.

In his Mathematical Psychics [5], Edgeworth presented a remarkable study of the exchanges of two commodities that might arise in an economy with two types of consumers. The first case that he considers concerns two individuals each of whom initially possesses certain quantities of each commodity. The result of trading consists of a reallocation of the total amounts of the two commodities and may, therefore, be described geometrically by a point in the Edgeworth box corresponding to that economy.

Edgeworth confines his attention to those exchanges which are Pareto optimal, i.e., those which cannot yield greater satisfaction for one consumer without impairing that of the other by means of additional trade. He further restricts the admissible final allocations to those which are at least as desired by both consumers as the allocation prevailing before trading. Those allocations which are not ruled out by either of these considerations constitute the "contract curve."

A numerical evaluation of the “dead loss” associated with a nonoptimal situation (in the Pareto sense) of an economic system is sought. Use is made of the intrinsic price systems associated with optimal situations of whose existence a noncalculus proof is given. A coefficient of resource utilization yielding measures of the efficiency of the economy is introduced. The treatment is based on vector-set properties in the commodity space.

Introduction

The activity of the economic system we study can be viewed as the transformation by n production units and the consumption by m consumption units of / commodities (the quantities of which may or may not be perfectly divisible). Each consumption unit, say the ith one, is assumed to have a preference ordering of its possible consumptions, and therefore an index of its satisfaction, si. Each production unit has a set of possibilities (depending, for example, on technological knowledge) defined independently of the limitation of physical resources and of conditions in the consumption sector. Finally, the total net consumption of all consumption units and all production units for each commodity must be at most equal to the available quantity of this commodity.

A mathematical model which attempts to explain economic equilibrium must have a nonempty set of solutions. One would also wish the solution to be unique. This uniqueness property, however, has been obtained only under strong assumptions,and, as we will emphasize below, economies with multiple equilibria must be allowed for. Such economies still seem to provide a satisfactory explanation of equilibrium as well as a satisfactory foundation for the study of stability provided that all the equilibria of the economy are locally unique. But if the set of equilibria is compact (a common situation), local uniqueness is equivalent to finiteness. One is thus led to investigate conditions under which an economy has a finite set of equilibria.

Now nonpathological examples of economies with infinitely many equilibria can easily be constructed in the case of pure exchange of two commodities between two consumers. Therefore one can at best prove that outside a small subset of the space of economies, every economy has a finite set of equilibria. For the precise definition of "small" in this context, one might think of "null" with respect to an appropriate measure on the space of economies. Such a null set, however, could be dense in the space and a stricter definition is required. Our main result asserts that, under assumptions we will shortly make explicit, outside a null closed subset of the space of economies, every economy has a finite set of equilibria.

A. Wald has presented a model of production and a model of exchange and proofs of the existence of an equilibrium for each of them. Here proofs of the existence of an equilibrium are given for an integrated model of production, exchange and consumption. In addition the assumptions made on the technologies of producers and the tastes of consumers are significantly weaker than Wald's. Finally a simplification of the structure of the proofs has been made possible through use of the concept of an abstract economy, a generalization of that of a game.

Introduction

L. Walras [24] first formulated the state of the economic system at any point of time as the solution of a system of simultaneous equations representing the demand for goods by consumers, the supply of goods by producers, and the equilibrium condition that supply equal demand on every market. It was assumed that each consumer acts so as to maximize his utility, each producer acts so as to maximize his profit, and perfect competition prevails, in the sense that each producer and consumer regards the prices paid and received as independent of his own choices. Walras did not, however, give any conclusive arguments to show that the equations, as given, have a solution.

Introduction

In the study of the existence of an equilibrium for a private ownership economy, one meets with the basic mathematical difficulty that the demand correspondence of a consumer may not be upper semicontinuous when his wealth equals the minimum compatible with his consumption set.One can prevent this minimum-wealth situation from ever arising by suitable assumptions on the economy; for example, in K. J. Arrow and G. Debreu [1], Theorem I, it is postulated that free disposal prevails and that every consumer can dispose of a positive quantity of every commodity from his resources and still have a possible consumption. However, assumptions of this type have not been readily accepted on account of their strength, and this in spite of the simplicity that they give to the analysis. Thus A. Wald [11, (Section II)]; K. J. Arrow and G. Debreu [ 1, (Theorem II or II')]; L. W. McKenzie [7], [8], [9]; D. Gale [4]; H. Nikaido [10]; and W. Isard and D. J. Ostroff [5] permit the minimum-wealth situation to arise but introduce features of the economy that nevertheless insure the existence of an equilibrium.

The opening address on the occasion of an important retrospective of a great artist is not expected to include a detailed analysis of all exhibits. Rather, the artist and his work are set in their context in the history of art; those features that characterize his work are highlighted and related to earlier and contemporary contributions. This is what I shall try to do on the present occasion; the reprinting of twenty scientific papers of Gerard Debreu.

The striking feature that characterizes Debreu's scientific contributions is that they are both general and simple - general in the sense of universal, in contrast to ad hoc specific or particular, simple certainly not in the sense of elementary, facile, or effortless, but in the sense of pure in contrast to compound and complex. To sense the depth of Debreu's contributions, one has to understand them in full generality.

Introduction

The question of the representation of a convex preference preorder by a concave utility function was first raised and answered by de Finetti (1949), and further studied by Fenchel (1953, 1956), Moulin (1974), and by Kannai in the forthcoming article “Concavifiability and constructions of concave utility functions” which also discusses the problem of least concave utility functions. To illustrate the value of such a concave representation by one example, we consider an exchange economy ℰ whose consumers have convex preferences, and, following Scarf (1967), we associate with the economy ℰ a game without side payments in coalition form. If the preferences of each consumer are represented by a concave utility function, then the characteristic set of utility vectors of each coalition is convex, as in the original definition of Aumann-Peleg (1960). The convexity of these characteristic sets permits, for instance, a simplification [Scarf (1965) and Ekeland (1974); see also the related article of Shapley (1969)] of the proof of the non-emptiness of the core of Scarf (1967).

By a simple reinterpretation of the concept of a commodity, the classical economic theories of equilibrium and optimality can be extended, without change of form, to the case where uncertain events determine the consumption sets, the production sets, and the resources of the economy. According to the usual definition, a commodity is a good or service whose physical characteristics and date and place of delivery, are specified; in this article, the definition of a commodity also specifies an exogenous event (which will be known to have, or not to have, occurred by the delivery date). By agreement of the contracting parties, delivery of the commodity is conditional upon the occurrence of this event. This extended definition, which originated in the paper by K. J. Arrow at the May 1952 CNRS Colloquium on Risk, permits an immediate transposition of the results of economics under certainty to those of economics under uncertainty, thanks to the identity of forms already mentioned; it also has the advantage of leading to a theory of uncertainty that makes no reference to the notion of probability. Admittedly, there are no markets for the commodities as defined here, but economists are familiar with the fruitful hypothesis that there exist futures markets for all commodities, although only an insignificant number of such markets are observed.

The observed state of an economy can be viewed as an equilibrium resulting from the interaction of a large number of agents with partially conflicting interests. Taking this viewpoint, exactly one hundred years ago, Léon Walras presented in his Eléments d'Economie Politique Pure the first general mathematical analysis of this equilibrium problem. During the last four decades, Walrasian theory has given rise to several developments that required the use of basic concepts and results borrowed from diverse branches of mathematics. In this article, I propose to review four of them.

The existence of economic equilibria

As soon as an equilibrium state is defined for a model of an economy, the fundamental question of its existence is raised. The first solution of this problem was provided by A. Wald (1935-1936), and after a twenty-year interruption, research by a large number of authors has steadily extended the framework in which the existence of an equilibrium can be established. Although no work was done on the problem of existence of a Walrasian equilibrium from the early thirties to the early fifties, several contributions, which, later on, were to play a major role in the study of that problem, were made in related areas during that period. One of them was a lemma proved by J. von Neumann (1937) in connection with his model of economic growth.

For an economic system with given technological and resource limitations, individual needs and tastes, a valuation equilibrium with respect to a set of prices is a state where no consumer can make himself better off without spending more, and no producer can make a larger profit; a Pareto optimum is a state where no consumer can be made better off without making another consumer worse off. Theorem 1 gives conditions under which a valuation equilibrium is a Pareto optimum. Theorem 2, in conjunction with the Remark, gives conditions under which a Pareto optimum is a valuation equilibrium. The contents of both theorems (in particular that of the first one) are old beliefs in economics. Arrow and Debreu have recently treated this question with techniques permitting proofs. A synthesis of their papers is made here. Their assumptions are weakened in several respects; in particular, their results are extended from finite dimensional to general linear spaces. This extension yields as a possible immediate application a solution of the problem of infinite time horizon (see sec. 6). Its main interest, however, may be that by forcing one to a greater generality it brings out with greater clarity and simplicity the basic concepts of the analysis and its logical structure.

The core of a finite economy has been shown to converge, as the number of its agents tends to infinity, under conditions of increasing generality in a series of contributions, of which the first, by Edgeworth (1881), studied replicated exchange economies with two commodities and two types of agents, and the latest, by Hildenbrand (1974), considers sequences of finite exchange economies (with a given finite number of commodities) whose distributions on the space of agents' characteristics converge weakly.However, information on the rate of convergence of the core seems to be contained in only two articles. In Shapley and Shubik (1969, section 5) an example is given of an Edgeworth replicated economy whose core converges like the inverse of the number of agents. Recently, Shapley (1975) provided examples of Edgeworth replicated economies whose cores converge arbitrarily slowly, but concluded with the conjecture that for any fixed concave utility functions only a set of initial allocations of measure zero will yield cores that converge more slowly than the inverse of the number of agents. The theorem stated below for replicated economies with arbitrary numbers of commodities, and of types, asserts that such is indeed the case provided that preference relations are of class C2, and satisfy the conditions listed in the definition of the economy. At the same time, the theorem implies that the set of exceptional allocations is closed as well as of measure zero.

The recent introduction of differential topology into economics was brought about by the study of several basic questions that arise in any mathematical theory of a social system centered on a concept of equilibrium. The purpose of this paper is to present a detailed discussion of two of those questions, and then to make a rapid survey of some related developments of the last five years.

Let e be a complete mathematical description of the economy to be studied (e.g., for an exchange economy, e might be a list of the demand functions and of the initial endowments of the consumers). Assumptions made a priori about e (e.g., assumptions of continuity on the demand functions) define the space ℰ of economies to which the study is restricted. By a state of an economy we mean a list of specific values of all the relevant endogenous variables (e.g., prices and quantities of all the commodities consumed by the various consumers). We denote by S the set of conceivable states. Now a given equilibrium theory associates with each economy e in ℰ, the set E(e) of equilibrium states of e, a subset of S (see Figure 1).

Part II is concerned solely with the application of the methodology developed in Part I to selected theoretical discussions. The static, simultaneous relations model is used in Chapter 8 to distinguish between political structure and political system by thinking of the former as a system of simultaneous relations and the latter as a solution of it. In Chapter 9 a planner's view of the making of sequential decisions by society's decision-making units is modeled with a system of periodic relations. Time paths are examined with respect to the logic of their potential for manipulation by the planner. Chapter 10 again utilizes a system of simultaneous relations to describe the general structure of society. Society itself is defined as a solution of the system dependent on certain parameter values. Change over time also occurs as parameters modulate according to specific periodic relations.

It should be emphasized that in all three cases, claims of originality are strictly limited. No outstanding problem is solved, no conclusion is extended, no new proposition is proved, and no new question is raised in any of the fields that these chapters represent. In fact, practically all approaches, concepts, hypotheses, and propositions are lifted directly from the references cited. Thus the models presented are at least as deficient as the original sources. Nevertheless, the contribution of these chapters is to show how the methodology of Part I can be applied to existing ideas. The result is an infusion of fresh rigor, precision, and mathematical-type argument into work that has already been completed.