Knowledge that Transforms

[The purpose of this paper is to prove an impossibility theorem for the existence of a social welfare function. Our treatment of the social welfare function differs from that of Arrow insofar as the social choice function generated through the social welfare function from the individual choice functions is defined in the disaggregated social state in his case, while in our case it is defined in the aggregated social state. We assume that every individual's choice function is defined in an n-dimensional commodity space. Then, the social choice function is defined in Arrow's case in an (m × n)-dimensional commodity space while it is defined in our case in an n-dimensional space. With our modification, the requirement insuring the existence of the social welfare function yielding social indifference surfaces becomes weaker than in Arrow's case. Nevertheless, we still obtain an impossibility.]

A new approach to the choice of econometric estimators, called small-sigma asymptotics, is introduced and applied to the choice of k-class estimators of the parameters of a single equation in a system of linear simultaneous stochastic equations. I find that when the degree of overidentification is no more than six, the two stage least squares estimator uniformly dominates the limited information maximum likelihood estimator in a certain sense. The small sigma method can be used on many problems in statistics and econometrics. THE STUDY OF simultaneous equation econometric models has led to many alternative estimators to ordinary least squares: single equation limited information maximum likelihood, and two stage least squares, for example. The behavior of these estimators has been difficult to describe, however, and it has been difficult to choose among these estimators. The work described in this paper explores this problem for the case in which lagged dependent variables are not permitted. To be most useful for normative purposes, a description must be detailed enough to give a good approximation and expose differences between estimators, and yet be simple enough to strengthen intuition and yield easily described comparisons. Since detail and simplicity are in conflict, approaches may differ in this respect. This paper introduces a new approach, based on asymptotic series in a scalar multiple, a, of the variance of the disturbance in the model. As a -+ 0 the regression function is an increasingly good description of the random variables generated. Intuitively this is suggested by Gauss' Theory of Errors the errors were never intended to be so as to swamp the regression function. One important approach used in the past is sample asymptotic theory. This reveals a persistent bias in ordinary least squares, and a sample asymptotic equivalence between two stage least squares and single equation limited information maximum likelihood. Additionally, Nagar [13] found the 11T term in the sample asymptotic bias and the 1/T and I/T2 terms of the moment matrix of two stage least squares. Economists have been uneasy, however, about application of sample theory to samples which may not be large in the relevant sense. Additionally sample asymptotic results often depend on an assumption about the asymptotic behavior of the moment matrix of exogenous variables which is difficult to justify.

[A general definition of majority decision in terms of a hierarchy of voting councils has been given by Murakami [3, 4]. The present article establishes a set of necessary and sufficient conditions for Murakami's majority decision or representative system in terms of properties of a group decision function for two alternatives. One corollary of the general theorem is Murakami's conjecture, which says that if a group decision function is dual, strongly monotonic, and nondictatorial, then it is a representative system.]

[In this study, a dynamic initial investment-borrowing model involving nonconvex investment effects and borrowing limitations is formulated as a discrete-time control problem. In the model, the firm's objective is to maximize, subject to constraints, the net worth of the firm over a finite decision-making period. Initial investment and borrowing are control parameters; and the scale of capacity use is the control variable. Investment costs, which reflect the "six-tenths" rule in particular, are nonconvex. Special considerations are thus involved in deriving the investment and borrowing rules. It is shown that the optimum must be at one of the following three points: (i) no investment and no borrowing, (ii) investment of just the endowment, and (iii) investment of the maximum amount possible. This result is especially important computationally, because the problem is convex at the points described by (ii) and (iii), and trivial at the origin. Therefore, the optimum may be computed by the use of published algorithms.]

[This note examines carefully the basic problems of excess demand functions that appeared in the paper "On the Stability of the Competitive Equilibrium, II" by K. J. Arrow, H. D. Block, and Leonid Hurwicz [1]. It attempts to show that nearly the same results can be obtained after some modifications of the assumptions and the methods of proof of lemmas.]

[This paper determines the time series behavior of investment, output, and prices in a competitive industry with a stochastic demand. It is shown, first, that the equilibrium development for the industry solves a particular dynamic programming problem (maximization of "consumer surplus"). This problem is then studied to determine the characteristics of the equilibrium paths.]