Knowledge that Transforms

This paper presents the classical theorem on the existence of equilibrium as it was proved in the 1950's with the various improvements that have been made since then.In particular, the elimination of the survival assumption and of the requirement of transitive preferences are carried through with a proof that uses a mapping of social demand.This approach favors intuitive understanding and generalization of the results.Finally, the role of the firm and the introduction of external economies are critically viewed. MYPURPOSE IS TO DISCUSS the present status of the classical theorem on existence of competitive equilibrium that was proved in various guises in the 1950's by Arrow and Debreu [1], Debreu [5, 6], Gale [8], Kuhn [14], McKenzie [17, 18, 19], and Nikaido [22].The earliest papers were those of Arrow and Debreu, and McKenzie, both of which were presented to the Econometric Society at its Chicago meeting in December, 1952.They were written independently.The paper of Nikaido was also written independently of the other papers but delayed in publication.The major predecessors of the papers of the fifties were the papers of Abraham Wald [31, 32] and John von Neumann [30], all of which were delivered to Karl Menger's Colloquium in Vienna during the 1930's.The paper of von Neumann was not concerned with competitive equilibrium in the classical sense but with a program of maximal balanced growth in a closed production model.However, he first used a fixed point theorem for an existence argument in economics and provided the generalization of the Brouwer theorem that was the major mathematical tool in the classical proofs.Wald achieved the first success with the general problem of the existence of a meaningful solution to the Walrasian system of equations.The proofs which were published used an assumption that later became known as the Weak Axiom of Revealed Preference.This axiom virtually reduces the set of consumers to one person, since it is equivalent to consistent choices under budget constraints.In a one consumer economy the existence of the equilibrium becomes a simple maximum problem and advanced methods are not needed.When many consumers with independent preference orders are present, it has been shown (Uzawa [29]) that fixed point methods are necessary.Wald also wrote a third paper whose main theorem was announced in a summary article [33], but which never reached

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The last section of this paper presents a rigorous version of Tiebout's theory of local public goods. It is shown that equilibria exist and are Pareto optimal. This rigorous theory follows closely the more rigorous part of Tiebout's work. This rigorous theory makes a number of very special assumptions which make local public goods essentially private. The body of this paper presents a series of examples, which show that if one tries to generalize the rigorous version of Tiebout's theory in a number of interesting directions, then equilibria may no longer exist or may not be Pareto optimal. The conclusion is that Tiebout's idea does not lead to a satisfactory general theory of local public goods. THE GOAL OF THIS PAPER is to point out that Tiebout's notion of equilibrium with local governments does not have the nice properties of general competitive equilibrium, except under very restrictive assumptions. Tiebout [39] suggested that there are competitive forces which tend to make local governments allocate resources in a Pareto optimal fashion. Consumers choose to live in those towns with the mix of taxes and public goods they prefer. Local governments choose this mix so as to attract inhabitants. This idea may seem intriguing, for it suggests that the invisible hand solves an important part of Samuelson's perplexing public goods problem [32]. Tiebout, in fact, makes an argument which is nearly rigorous. I give a rigorous version of his argument at the end of the paper. However in this rigorous version, so many restrictive assumptions are made that public goods become essentially private. In the body of the paper, I give a series of examples with which I try to convince the reader that one is forced to adopt Tiebout's restrictive assumptions. The idea is that if one changes any of his assumptions, then either equilibria may not exist or may not be Pareto optimal. My examples are presented in the context of a general class of Tiebout models. I consider several subclasses, one of which is the special case considered by Tiebout. In each of the subclasses except that considered by Tiebout, I give a counterexample either to the existence of equilibrium or to its Pareto optimality. The subclasses are so chosen that the difficulties they reveal would be shared by any reasonable Tiebout model which differed from his special case. I believe that my examples controvert Tiebout's suggestion [39, last paragraph] that his theory compares favorably with competitive equilibrium theory. Most of the examples in this paper have already appeared in the literature. I cite related work as I go along. What is new here is that I assemble the examples in a unified argument.

In this paper [the authors discuss] five procedures for incorporating demographic variables into theoretically plausible demand systems: translating scaling and the Gorman reverse Gorman and implicit Prais-Houthakker procedures.... These five procedures are used to incorporate a single demographic variable--the number of children in a household--into the generalized CES demand system using household budget data for the United Kingdom for the period 1966-1972 (EXCERPT)

The bulk of this work was completed at the Rand Corporation and several extensions were made while I was on leave to the Department of

[The theory of duality has been an extremely useful tool in the analysis of the standard models of consumer and producer behavior. This paper describes an extension of the theory to a wider class of problems of static optimization. The generalized duality theory is then applied to the integrability question in fairly general optimization models. A major gap in the comparative statics of optimization models is partially closed.]

Ce modele est principalement utilise en analyse input-output: A represente alors les echanges interindustriels de l'annee de base, u et v les consommations intermediaires totales par produit et par branche de l'annee de projection; plus generalement on peut appliquer cette methode pour desagreger des projections globales de flux positifs (importations par pays et produit, flux de transports . . ) lorsque l'on suppose une certaine stabilite structurelle. Diverses interpretations economiques des coefficients r, et s1 (qui sont en fait strictement positifs puisque u, et v; le sont aussi) ont ete avancees;1 notons toutefois que les grandeurs absolues de r et s n'ont pas de signification intrinseque, car si (r, s) est