modities (such as various consumer goods and labor), M fixed factors (such as land, natural resources and various types of fixed capital), and a government which taxes commodities and fixed factors in order to finance various govern- ment expenditures. It is well known2 that if the government can raise its required revenue by taxing the fixed factors alone, then the resulting allocation of resources is Pareto optimal-no single household's utility or real income can be increased without decreasing the utility of some other household. Suppose we are at an initial equilibrium where government revenue is being raised by taxing the fixed factors alone. Then the resulting equilibrium can be rationalized by maximizing a certain weighted sum of utility functions subject to various feasibility constraints. Now think of the government replacing the taxes on fixed factors with distortionary commodity taxes. In Section 3, we calculate the second order directional derivative of the above weighted sum of utility functions with respect to any feasible direction of tax change, evaluated at the initial equilibrium which is Pareto optimal. Of course, the first order directional derivatives of the weighted sum of utility functions with respect to feasible directions of tax change are zero evaluated at this initial equilibrium. We obtain a measure of economic due to tax distortions which is virtually identical to that of Boiteux (3, p. 113) and which bears a resemblance to the dead loss of Hotelling (22, p. 254), the consumer's surplus measures of Hicks (19; 20, pp. 330-3), and the deadweight loss measure of Harberger (16, p. 61; 17, p. 788). In Section 4, we calculate a measure of welfare based on Debreu's (4, 5) coefficient of resource utilization (which is a modification of a measure of due to Allais (1, 2)) and we show that under certain conditions, the Hotelling,

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

For more than three decades, economic columnist Joseph A. Livingston has canvassed a panel of economists twice a year, eliciting their six-month and twelve-month forecasts for more than a dozen key variables. This study analyzes whether experts' predictions are unbiased, and whether complete use was made of all relevant, known information (unbiasedness and completeness being necessary conditions for fully rational expectations). Little bias was found in either half-year or full-year predictions, but extensive underutilization of information-particularly data on monetary growth-occurred. To prophecy is extremely difficult-especially with respect to future. Chinese proverb Do ECONOMISTS' EXPECTATIONS regarding key price and nonprice variables utilize all known, relevant information, in an unbiased, efficient manner? This is a worthy subject for research, for several reasons. Properties of experts' predictions likely form an upper bound for those of laymen. Further, as John Muth [14] has noted, the character of dynamic processes is typically very sensitive to way expectations are influenced by actual course of (p. 316); hence, we need to know precisely how events do affect expectations. Finally, common practice of replacing a variable's (generally unobserved) expectation with a proxy based on its past values will be unbiased (and will not cause bias in other

and where tl, 2, ... , n-k is a set of independent standard normal variates. The significance points of dM, dL, and du may be evaluated by the methods of Imhof [3] and Pan [8]. Savin and White [9] have noted that Koerts and Abrahamse's [5, pp. 159-160] FORTRAN implementation of Imhof's procedure fails to converge when the significance point is near A1. This defect is easily remedied by replacing Imhof's lower bound for the inverse of the truncation error by