SUMMARY THE TECHNIQUES of linear programming are here explained in a commercial application-blending aviation gasolines. Blending is critically important to almost all other areas of programming in an integrated oil company. Intelligent programming of production, transportation, manufacturing, or marketing generally requires solution of blending problems as an initial or integral part of the whole process for it is in blending that the final outputs are determined. In the first part of the paper, questions of optimum programming in a given technological and institutional structure are explored. Computations are executed primarily by means of the simplex technique of Dantzig [4]. Because of the presence of multiple degeneracy and the absence of a general method (at the time these computations were made)2 of handling such problems, it was not possible to rest securely on the simplex method. Alternative methods of computation were, therefore, explored and bounding techniques were employed to discover possible divergences from optimality. Finally, to test the validity of the results, calculations undertaken by a company programming official were obtained for purposes of check and comparison. A relatively simple program is first calculated and more than one optimal program obtained. These results are then extended to more complex problems. Finally, the sensitivity of the matrix to possible changes in the coefficients is studied. The problem of linear programming may be sta-ted as follows: A

This paper examines the identifiability of the coefficients of a single equation in a simultaneous equation model which is nonlinear only in the variables. The concept of identifiability in this model is motivated and developed using the closely related concept of observational equivalence. This framework is then utilized to develop necessary and sufficient conditions for identifiability when the disturbances are required to be independent of the exogenous variables. The approach recommended by Fisher is shown to yield sufficient but not necessary conditions for identifiability. For several relatively common special cases the necessary and sufficient conditions are found to simplify to the familiar rank condition for identifiability in the linear model. THE SIMULTANEOUS EQUATION MODEL that is nonlinear only in the variables has enjoyed widespread application in economics. Such models are linear in the parameters and typically seem to be linear in the variables as well, when viewing a single equation. In many models the nonlinearity in the variables arises due to endogenous variables entering in different forms in different equations (logged and unlogged form, for example). In macroeconometric models nonlinearity in the variables arises when the model includes endogenous real, nominal, and price variables, which are nonlinearly related.2 Whatever the reason for the nonlinearity in the variables, it is important to determine the conditions under which the equations of such models can be identified.

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 the regression equation C = A,B + e where A is a p x q stochastic matrix whose elements are independently distributed and contemporaneously correlated with the elements of 8, the lth moment of the least squares estimator of (3 exists if and only if l < p - q + 1. In particular, this implies that the lth moment of the k-class estimator of the coefficients of the G1 -1 non-normalizing endogenous variables of an equation with K1 included and K2 excluded exogenous variables in a simultaneous system with N observations exists if and only if 1 < M where