Knowledge that Transforms

[This paper discusses in general the use of a priori information in regression analysis and explores in detail the use of mixtures of normal prior distributions. It includes an application to distributed lag analysis.]

[An exact discrete model is derived from a recursive model consisting of a set of rth order stochastic linear differential equations with constant coefficients such that observations generated at equidistant points of time by the differential system satisfy the discrete model irrespective of the length of sampling interval. The difficulty of estimating the exact model subject to a priori restrictions makes it necessary to approximate the differential system by a non-recursive discrete model that maintains the structural form. This discrete model has a moving average disturbance term of order r - 1, but the co-variance function of this process is approximated by a function that is independent of the parameters of the continuous model. The eigenvalues and eigenvectors of the approximations to the differential system as well as their asymptotic variance matrices are also derived but, like the approximate asymptotic variance matrices of the parameter estimates, these variances are about biased probability limit]

1. TO BE CLEAR from the beginning, this note does not provide practical aids to identification; the information is very unlikely to be available. Rather, this paper serves an expository purpose, that of seeking to clarify two ideas: first, multiple-identification, by presenting and analyzing an example within a linear system, and second, Working's and Fisher's ideas on relative variances as identification aids, by showing the consequences of knowing that one variance is a specific large multiple of another. Along the way we provide a correction to the econometrics literature.

[We shall present an alternative proof of G. Debreu's theorem on the finiteness of the set of equilibria [1], and some related new results. The set of economies with finitely many equilibria is dense with respect to a very weak topology on the set of exchange economies. The equilibrium correspondence and the number of equilibria are continuous with respect to a strong topology on the set of regular economies where "regular" is defined as in [1]. Perhaps the methods used in this paper are more interesting than the results. The key concept we use is that of transversality. The essential regularity property of regular economies is that their excess demand functions are transversal to zero.]

IN THE GROWTH model of Kemeny-Morgenstern-Thompson [2, pp. 115-135], the production space is a closed convex and polyhedral cone in R2, with some further properties. In the growth model of Gale [1, pp. 285-303], the production space has to fulfil the same assumptions except that the cone need not be polyhedral. Therefore the model of Gale can be regarded as a generalization of the KMT-model. In this paper it will be shown that in contradiction to a central theorem of Gale, the existence of an optimal price vector cannot be guaranteed in this generalized production space. To describe the difficulty in the growth model of Gale, we enumerate the properties of the production space and the basic definitions given by him. The technological possibilities of production are described by the production space

The paper discusses the allocation of output among consumption and two types of capital with different gestation periods. Along an optimal path, we show that the imputed prices of capital goods, from the time they start production, do not exceed the prices of output, which are not less than the marginal instantaneous utility of consumption. A simple numerical example helps to illustrate some further implications of the model. RECENT PAPERS on optimal growth consider models of allocation of resources between consumption and investment. It is invariably assumed that investment results in an instantaneous increase in the stock of capital. Such assumptions obscure differences in the gestation periods among various capital goods. In [1] we discuss how a growth problem with time lags can be formulated and interpreted and explain the derivation of the necessary conditions for optimization. In this note we study the effects of differences in gestation periods on optimal investment plans for a growth problem including depreciation and population growth. Consider an economy where two capital goods and labor are used in the production of a single commodity. The per capita output at time t is given by the production function: f (kl, k2), where k, is the per capita stock of capital of type one, and k2 is the per capita stock of capital of type two. From now on, all variables will be per capita and we drop the designation. We assume the following:

[This paper is concerned with the introduction of input or output targets in decentralized planning processes. The process which is studied is that of Dantzig-Wolf-Malinvaud. All the properties of a "good" scheme are present when the central board designates a neighborhood of a point of the production set (instead of an exact output target) where each firm or sector is constrained to maximize an index.]