This paper formulates the notion of stochastic equilibria as invariant probability distributions consistent with the behavior patterns of individuals and the disequilibrium adjustment mechanism of the economy. Conditions for existence, uniqueness, and stability of such equilibria are examined. WE CONSIDER A CLASS of problems in this paper in which the economic environment is stochastic. We will be concerned primarily with developing an equilibrium concept for general equilibrium models of this type. However the essential ideas can be carried over directly to partial equilibrium applications. The choice of the specific general equilibrium model used results primarily from a desire to facilitate comparisons with earlier work on alternative equilibrium concepts for this model (see Hildenbrand [9] and Majumdar and Bhattacharya [2 and 3]). Randomness can arise from several sources. We will be considering, for concreteness, a simple exchange economy in which the basic data are the preferences and endowments of the economic agents. Either of these can be random. Typically, randomness of endowments can be allowed for by creating contingent markets in which case the Arrow-Debreu deterministic equilibrium suffices. It is conceptually much more difficult to create markets contingent on tastes due to the difficulties of discovering the true taste pattern of an individual, difficulties which do not arise in the case of endowment vectors which can be observed directly. We will be considering an economy without markets for every future contingency and thus there will remain some randomness. This residual uncertainty in the economy necessitates equilibrium concepts other than the Arrow-Debreu system of market clearing prices. 2. NOTATION

Under the assumption of single peaked preferences, the majority rule equilibrium considered as a correspondence from the voters' preference is shown to be continuous. We also complement the work of Fishburn [6], who first presented a general location theorem for majority rule equilibriums, by dropping the assumptions that the alternative set is finite and that voters' preferences are strict partial orders. SINCE THE WORK of Black [3] on simple majorities and single-peaked preferences, much work has been done in deriving conditions for which some state achieves a majority over all other states. It is also well known that the equilibrium in many cases is the median of the distribution of most preferred states of the voters. (See [3, pp. 14-18].) We present here a result concerning the continuity of the majority rule equilibria. Specifically, we show that in the case of single-peaked preferences, the majority rule equilibrium depends only on the peaks of the voters' preferences and not on the transitivity properties of these preferences. We then show that the equilibrium, viewed as a correspondence of these peaks, is continuous (i.e., both upper and lower semicontinuous). This result is especially important when one tries to prove the existence of an equilibrium in a political-economic system. Most existence theorems are based on fixed point theorems which require at least upper semicontinuity of the correspondences being studied. The theorem proved here shows that the majority rule equilibrium is continuous in voters' peaks. If these are in turn continuous functions of other parameters such as prices, then our result might aid one in deriving general existence theorems for social equilibria. (See [3 or 4].) The only related work seems to be that of Kelly [10] who investigates the existence of a continuous numerical representation of the social preference relation generated by majority rule. He shows that one cannot expect continuity even if the social preference relation has other properties, such as transitivity. Specifically, he presents a case in which a majority of voters are indifferent between two points. In a neighborhood of one of these points, this majority rules between points in the neighborhood and the other point, but a minority (since the majority is indifferent) rules between the two given points. Hence one cannot expect continuity of the

[In this paper we analyze implications of a singular contemporaneous disturbance covariance matrix for the estimation and hypothesis testing of systems of equations with autoregressive disturbances. We find that this singularity imposes restrictions on the parameters of the autoregressive process. When these restrictions are not imposed, the specification, maximum likelihood estimates, and likelihood ratio test statistics are conditional on the equation deleted. Furthermore, singularity of the contemporaneous disturbance covariance matrix raises issues concerning identification of parameters of the autoregressive process. This identification problem complicates the interpretation of likelihood ratio statistics. The above results are illustrated with an empirical example.]

[Models of the behavior of populations of self-reproducible natural resources in an economic framework have rarely anticipated the consequences of different forms of production functions. This paper investigates sufficient conditions for extinction in a very general model as well as a model having a specific production function. In the second section additional considerations relating to extinction are deduced as well as the existence of a watershed level of population. These conclusions are exemplified using data from one particular population of red deer.]