Knowledge that Transforms

IT IS WELL KNOWN that least-squares estimates of the coefficients of a regression equation are inconsistent if any of the regressors are measured with error. The nature of these inconsistencies has been examined by Aigner [1], Blomqvist [2], Chow [3], Levi [5], McCallum [6], and Wickens [10] for the case in which a single regressor is subject to measurement error. The purpose of this study is to examine the nature of these inconsistencies when more than one variable is measured with error. We begin by reviewing the case of one variable measured with error, developing a unified treatment of issues which previously have been discussed separately. Concentrating on the case in which two regressors are measured with error, we then examine how the predictions of the one erroneously measured regressor model must be qualified when more than one regressor is subject to measurement error.

Here q E R n, the n-dimensional Euclidean space, x (t) is a R n-valued function, and u (x, t) Rn x [0, oo) -> R, is a nonnegative real function normalized such that u (0, t) = 0 for every t. Inequalities between vectors are understood coordinatewise. We shall be concerned mainly with the case where (L) might not have optimal solutions. We shall develop, characterize, and apply the notion of a generalized solution of this optimization problem. This variational problem arises in several contexts. See Aumann and Shapley [4, Chapter VI], Yaari [11], and the references therein. We shall adopt here the following interpretation. The coordinates of the vector q represent initial endowments of several exhaustible resources. The function x = x (t) is the rate at which the resources al-e consumed. The constraints mean that there is no re-filling (x (t) ? 0) and that the overall consumption is bounded by q. If the rate x (t) is consumed at t, it contributes the rate u(x(t), t) to the total utility. The latter is represented as the integral which has to be maximized. It is our economic interpretation that has led us to the choice of [0, xo) as the domain of integration. There is no mathematical significance in this choice. The analysis shows however that the behavior at t finite might be different from the behavior at t = oo; the difference is caused by the fact that a finite t has a neighborhood with finite Lebesgue measure, in contrast to t = 00, which has only neighborhoods of infinite Lebesgue measure. We shall refer to the role of this difference in the text.

[This note provides a rigorous description of a game in a strategic form whose Nash equilibria are all strong equilibriaccoinciding with the Walras equilibria of the underlying Arrow-Breau pure exchange economy.]

[The theory of monetary policy is examined as it pertains to the functions of money as intermediating intergenerational trade as well as providing a useful service return. Finite economic lives are shown to alter the characterization of the optimal inflation rate from that suggested by models with infinitely long lived agents. In uncertain environments with sequential trade, policy is examined as altering the range of possible trades between agents of successive generations. In some cases, fully anticipated activist feedback policies may increase expected utility from that attainable with passive policies.]

[Methods of estimation for markets in disequilibrium have been limited to the single-market case. However, the spill-over effects of the unsatisfied demand or supply in other markets are considered to be an essential feature of disequilibrium analysis. This paper (i) develops a two-market disequilibrium model that is amenable to estimation; (ii) provides the maximum likelihood method and the two-stage least squares method for estimation; and (iii) generalizes this disequilibrium model to the n-market case, showing sufficient conditions for the existence and uniqueness of a quantity-constrained equilibrium--i.e., for its solvability as an econometric model.]

IN HIS SURVEY ON RATIONAL EXPECTATIONS, R. Shiller indicates that the difficulty of obtaining explicit solutions for linear difference models under rational expectations may have hindered their use [14, p. 27]. The present paper attempts to remedy that problem by giving the explicit solution for an important subclass of the model Shiller refers to as the general linear difference model. Section 1 presents the form of the model for which the solution is derived and shows how particular models can be put in this form. Section 2 gives the solution together with the conditions for existence and uniqueness. 1. THE MODEL The model is given by (la), (lb), and (1c) as follows:

[The purpose of this paper is to reconsider the theory of Pareto optimal redistribution from a game-theoretic point of view. We define an income redistribution game in strategic form, which may allow many varieties of utilityinterdependencies. The strategy of each individual is a vector that describes a plan of transfers from him to every other individual. The results state that while a Pareto optimal redistribution is always achieved by a Nash equilibrium in the two-person case, it is not so when there are more than two individuals. The sufficient condition we derived seems unlikely to be satisfied in general.]