Stationary Markov Equilibria

We establish conditions under which there are equilbria (in various settings) described by ergodic Markov processes with a Borel state space S. Let P(S) denote the probability measures on S, and let s - G(s) C:P(S) be a (possibly empty-valued) convex valued correspondence with closed graph characterizing intertemporal consistency, as prescribed by some particular model. A measurable set J C S is self-justified if G(s) n P(J) is not empty for all s E J. A Time-Homogeneous Markov Equilibrium (THME) for G is a self-justified set J and a measurable selection II : J - P(J) from the restriction of G to J. The paper gives sufficient conditions for existence of compact self-justified sets, and applies the theorem: If G has a compact self-justified set, then G has an THME with an ergodic measure. The applications are (i) stochastic overlapping generations equilibria, (ii) an ex- tension of the Lucas (1978) asset market equilibrium model to the case of heterogeneous agents, and (iii) equilibria for discounted stochastic games with uncountable state spaces. Darrell Duffie is with the Graduate School of Business, Stanford University; John Geanakoplos is with the Department of Economics, Yale University; Andreu Mas-Colell is with the Department of Economics, Harvard University; and Andrew McLennan is with the Department of Economics, University of Minnesota. We are grateful for comments from Larry Blume, Hugo Hopenhayn, Sylvain Sorin, Jean- Francois Mertens, Richard Torres. and Larry Jones.