THE STABILITY PROPERTIES of a competitive economy are well known-at least under the artificial tatonnement assumption which avoids the problem of who changes prices. This note demonstrates that similar, if not stronger, properties carry over to a monopolistic economywhere price formation is completely explained by the independent optimizing behavior of individual agents. The discussion may be interpreted as that of monopoly as the opposite polar case to that of competition or, equivalently, as that of the stability of the concept of equilibrium presented in an earlier paper [1]. Price adjustment is by individuals, following [3 and 4], through their perceived demand curves, following [2 and 5]. There are n commodities (the last being a numeraire) identified by ownership: agent i is endowed with one unit of commodity i and no other commodity. Each agent (i) chooses a price pi ) 0 for his commodity and a consumption x' e R% , so that a state of the economy is an array (p1. p, x . x), or (p, X). Given some existing state (p, X) each agent (i) first chooses pi to maximize his expected income pixi, where xi is his expected sales, given by his perceived linear demand curve (with negative slope one, say) xi(pi) = x-i + Pi - pi; here Xi = I 5i is the existing aggregate demand. Taking account of the constraint xi < 1 this gives

[Previous work on non-tâtonnement processes has allowed only trading of titles to commodities or promises to produce to take place out of equilibrium. The present work allows production and consumption to take place. The basic model used is that of the Hahn Process, since the making of irreversible commitments in production and consumption seems especially suited to a model whose basic feature is the decline of target profits and utilities. The attempt to introduce production and consumption out of equilibrium brings to the fore a number of problems implicit in most stability analysis which must now be explicitly faced.]

PROBLEMS OF COMPARING or choosing among models of a stochastic process are frequently encountered in empirical research. In many such situations, conventional statistical procedures offer little guidance since they assume that the model is given. If the alternative models can be nested in a more general model, standard estimation and testing procedures can be employed. Often, however, such general models are not readily available and other considerations may dictate against their use. Recently, there has been considerable progress in the development of methods for comparing alternative non-rested models. A review of this work both Bayesian and non-Bayesian, is given in Gaver and Geisel [1]. Discussions of the Bayesian approach to the comparison of linear regression models are given in Zellner [3, Ch. 10] and Lempers [2], among others. In this paper we consider Bayesian comparison of linear models in which the disturbances have non-scalar covariance matrices. General posterior odds expressions are given and specialized to the case of first order autoregressive disturbances. We also consider a specification error problem in this context; that is, we examine the effect of ignoring the non-scalar covariance structure on the posterior odds ratio. For the first order autoregressive disturbance case we give an approximate expression indicating the magnitude of the error involved in computing the posterior odds ignoring the serial correlation. The accuracy of this approximation is investigated via a small sampling experiment. We use the following notatiori: Let the ith model, Mi (i = 1, 2,. .., N), be y = Xi/3i + yi where y is a T x 1 vector of observations on the random (dependent) variable of interest, Xi is a T x ki matrix of observations on the explanatory variables of Mi (Xi is assumed to be non-stochastic with rank ki), p3i is a ki x 1 vector of unknown parameters of Mi, and ui is a T x 1 vector of disturbances of Mi (yi is assumed to have a normal distribution with E(yi) = 0, and E(yiyii) = U22i where 2Ji is an unknown T x T positive definite symmetric matrix with trace (i) = T).2 Probability functions for the models are denoted by P( ), densities for parameters by 7r( ), and densities for observations by p( ).

and FORTRAN IV which is designed to accommodate most researchers' everyday econometric needs. However, this program is particularly useful when spectral methods are combined (in an ex post sense) with regression and simultaneous equations estimation., Residuals from regression or simultaneous equations estimation are easily saved by EAS and used in later spectral computations by using only two program statements.1 All spectral computations are highly efficient since the fast Fourier transform techniques developed by Cooley and Tukey [2] are used throughout. The program also allows algebraic expressions to be used directly in regression statements to define dependent or independent variables. Hence, special regression equations like the harmonic analysis model can be easily estimated with a single program statement. A partial enumeration of the program's capabilities is as follows: ordinary and weighted least squares regression; multivariate regression and estimation of Zellner's seemingly unrelated regression system; structural estimation of simultaneous equations by two-stage least squares, three-stage least squares, limited information maximum likelihood, k-class, double k-class, h-class, and Nagar's minimum bias k-class methods; power spectrum analysis, cross spectrum analysis, and simple frequency domain regression; random number generation and Monte Carlo methods; principal components analysis; estimation of partially nonlinear models by likelihood search techniques; and estimation of certain distributed lag models by Dhrymes' [3] methods. The program accommodates problems in which the sample size and number of yariables do not exceed 32,767 and 1,823, respectively, but the dynamic core allocation features of PL/1 are used to economize on all smaller problems. The design of the program is especially useful when a large data bank (subject to the abQve limitations) is to be maintained, updated, and periodically accessed for various types of econometric analyses. Most small- to medium-sized

[We suggest a frequency-domain class of instrumental variables estimators for all of part of an open, linear system of differential equations. While the estimators have similar statistical properties to those of [6] they seem preferable computationally in those situations in which they can be used. The estimation procedure consists of two basic steps, the first of which we describe as consistent and the second, efficient. We discuss the possible instruments that can be used. This type of procedure, like that of [6], could also be used to efficiently estimate discrete time systems, under weak conditions on the residuals.]

[This paper is concerned with the estimation of a system of simultaneous linear differential equations that involves predetermined variables. The system is replaced by a discrete approximation that is most conveniently handled in the frequency domain. Our method of estimation is nonlinear least squares. We state conditions under which the estimators will have asymptotically desirable properties. The most notable of these is an aliasing condition on the predetermined variables.]