Equilibria in stochastic economic models are often time series which fluctuate in complex ways. But it is sometimes possible to summarize the long run, characteristics of these fluctuations. For example, if the law of motion determined by economic interactions is Markovian and if the equilibrium time series converges in a specific probabilistic sense then the long run behavior is completely determined by an invariant probability distribution. This paper develops and unifies a number of results found in the probability literature which enable one to prove, under very general conditions, the existence of an invariant distribution and the convergence of the corresponding Markov process. VIRTUALLY ALL OF ECONOMIC THEORY focuses upon the study of economic equilibrium. This concept has recently undergone several subtle elaborations. No longer must a system of markets in equilibrium be thought of as one at rest in a static steady state. Instead there is a growing body of literature (e.g., [4, 5, 12, 16, 20, 21]) which defines equilibrium as a stochastic process of market clearing prices and quantities which is consistent with the self-interested behavior of economic agents. Needless to say equilibrium stochastic processes can be very complex time series which fluctuate in irregular ways. For theoretical and econometric purposes it is useful to have a convenient way of summarizing the average behavior of such processes over time. This paper draws together and unifies a number of fundamental results from the probability literature which enable one to do this for discrete time, Markov processes on general state spaces. The starting point of the analysis is a set S of economic states (e.g., prices and/or quantities). The only technical restriction placed upon S is that it be a Borel subset of a complete, separable metric space. The second datum is a transition probability P(s, ) on S. The number P(s,A) records the probability that the economic system moves from the state s to some state in the Borel subset A of S during one unit of elapsed time. In economic applications the transition probability is usually derived from hypotheses about market clearing and the maximizing behavior of economic agents. The transition probability (together with an initial probability measure on S) defines a discrete time Markov process. One way of summarizing the dynamic behavior implied by P is to look for an invariant probability. A probability measure X on S is invariant for P if for all Borel subsets A of S one has the equality f P(s, A )X(ds) = X(A). An invariant probability is a kind of probabilistic steady state for the dynamics defined by P. Of course there may be no invariant probability for P at all; and even if one exists it may convey no information about the behavior of the process over time except under very special initial conditions. There is a second way of summarizing the behavior of Markov processes defined by the transition probability P. Let P (s,A) denote the n step transition

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This paper establishes a natural and satisfying characterization of the class of collective choice rules which are acyclic and satisfy the Arrow axioms (unrestricted domain, independence of irrelevant alternatives, and the weak Pareto principle). We show that, when the number of alternatives is larger than the number of individuals, there must exist an individual who can at least some critical number of pairwise decisions. This critical number of veto pairs depends on the number of alternatives and individuals, and, as the number of alternatives increases without limit, the fraction of all pairs which some individual can veto approaches unity. We also present a global veto theorem and an axiomatic characterization of the Pareto extension rule which utilizes acyclicity rather than quasi-transitivity. ARROW [1] SHOWED that the only collective choice rules that yield weak order social preference relations and satisfy unrestricted domain, independence of irrelevant alternatives, and the weak Pareto principle are dictatorial. Gibbard [9] demonstrated that by relaxing the rationality requirement from transitivity to quasi-transitivity (i.e., transitivity of the strict preference relation) we can evade the letter though not the spirit of the Arrow dictatorship result: oligarchy, a weaker form of dictatorship, still obtains when the other three axioms are imposed. In this paper we prove a theorem parallel to those of Arrow and Gibbard for the weaker rationality requirement of acyclicity (i.e., the absence of cycles of strict preference). Since acyclicity is a necessary and sufficient condition for the existence of a nonempty set of maximal elements in every finite feasible set, there are powerful reasons for imposing it. Moreover, as we argue in Blair and Pollak [2], it is difficult to justify any stronger rationality property such as quasi-transitivity without at the same time justifying some even stronger rationality condition which implies dictatorship. Our principal result shows that, when the number of alternatives is larger than the number of individuals, there must exist an individual who can at least some critical number of pairwise decisions. (We say that individual i has a veto over the ordered pair (y, x) if he is weakly decisive for x against y-that is, if his strict preference for x over y implies weak social preference for x over y, regardless of the preferences of other individuals.) This critical number of veto pairs depends on the number of alternatives and the number of individuals. As the number of alternatives increases without limit, the fraction of all pairs that some individual can veto approaches unity. There may be more than one individual who can veto at least the critical number of pairs; indeed, it is possible for every individual to have a veto over every ordered pair of alternatives.

In this paper we analyze the solutions of linear econometric models with rational expectations. More precisely, we describe in detail the set of all the solutions; in particular this set is shown to be much larger than the sets previously considered. We also study various criteria of selection in this set of solutions and we examine to what extent these criteria redtiuce the set of the solutions.

[In the classical linear regression model the conflict between the W, LR, and LM tests is due to the tests not having the correct significance level. This paper shows that the probability of conflict can be substantial when the three tests are based on the asymptotic chi-square critical value. For this model some computable correction factors for the chi-square critical values are examined, including those derived from a second-order Edgeworth approximation to the exact distributions. It is shown that the probability of conflict between the Edgeworth size-corrected tests is of no practical importance over a wide range of conditions.]

[In a portfolio problem with given asset returns, the portfolio efficient set is the set of portfolios chosen by any risk averse agent. Using an approach of Peleg and Yaari [13], we characterize the portfolio efficient set and derive some of its properties. In particular, we show that it may not be convex, proving that a central result of mean variance theory, the efficiency of the market portfolio, does not generalize. Finally, a characterization of the efficiency of several observations gives a version of revealed preference theory for incomplete markets.]