Stationary Markov Equilibria

Abstract

We establish conditions which (in various settings) guarantee the existence of equilibria described by ergodic Markov processes with a Borel state space S. Let 9(S) denote the probability measures on S, and let s-G(s) c 4?(S) be a (possibly empty-valued) correspondence with closed graph characterizing intertemporal consistency, as prescribed by some particular model. A nonempty measurable set J c S is self-justified if G(s) n 9?(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 TI: J-9 _(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 is convex-valued and 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 extension of the Lucas (1978) asset market equilibrium mnodel to the case of heterogeneous agents, and (iii) equilibria for discounted stochastic games with uncountable state spaces.

Bayesian general equilibrium

“…In many applications, recursive equilibria are unique but only because the assumption of point expectations is made (price function) and distributions over exogenous variables are taken as given. If, in contrast, we use the concept of Markov equilibria defined as a joint Markov process over exogenous and endogenous variables (Duffie et al, 1994), then indeterminacy of equilibria is the rule. Krebs (1997) has shown how to use maximum entropy to resolve this indeterminacy issue by using the concept of statistical expectational equilibrium (SEE), which is in a sense the most likely rational expectations equilibrium.…”

Analysis of Numerical Errors

“…Follow-ing Duffie et al (1994), the existence of a Markov equilibrium in a generalized space of variables is proved in Kubler and Schmedders (2003) for an asset pricing model with collateral constraints. Feng et al (2012) extend these existence results to other economies, and define a Markov equilibrium as a solution over an expanded state of variables that includes the shadow values of investment.…”

“…Blume (1982) and Duffie et al (1994) follow (i), and are required to randomize over the existing equilibria to build a convex-valued correspondence. Randomizing over the equilibrium correspondence may result in an undesirable expansion of the equilibrium set.…”

Recursive equilibrium in endogenous growth models with incomplete markets

“…Thus, there is no stationary Markov equilibrium in the sense of Duffie et al (1994). However, the ratio variablesk (ratio of physical to human capital) andc (ratio of consumption to wealth) follow a stationary Markov process if the exogenous shock process {S t } is stationary.…”