Consumption dynamics in general equilibrium: A characterisation when markets are incomplete

We introduce a methodology for analysing infinite horizon economies with two agents, one good, and incomplete markets. We provide an example in which an agent’s equilibrium consumption is zero eventually with probability one even if she has correct beliefs and is marginally more patient. We then prove the following general result: if markets are effectively incomplete forever then on any equilibrium path on which some agent’s consumption is bounded away from zero eventually, the other agent’s consumption is zero eventually–so either some agent vanishes, in that she consumes zero eventually, or the consumption of both agents is arbitrarily close to zero infinitely often. Later we show that (a) for most economies in which individual endowments are finite state time homogeneous Markov processes, the consumption of an agent who has a uniformly positive endowment cannot converge to zero and (b) the possibility that an agent vanishes is a robust outcome since for a wide class of economies with incomplete markets, there are equilibria in which an agent’s consumption is zero eventually with probability one even though she has correct beliefs as in the example. In sharp contrast to the results in the case studied by Sandroni (2000) and Blume and Easley (2006) where markets are complete, our results show that when markets are incomplete not only can the more patient agent (or the one with more accurate beliefs) be eliminated but there are situations in which neither agent is eliminated. Journal of Economic Literature Classification Numbers: D52, D61 Correspondence To: Pablo F. Beker Department of Economics University of Warwick Coventry CV4 7AL UNITED KINGDOM Telephone: +44 2476 150588 Fax: +44 2476 523032 e-mail: [email protected] ∗ An earlier version of the paper was circulated with the title “Economic Survival when Markets are Incomplete”. We thank M.Quinzii and M. Woodford for very useful exchanges, D. Easley for comments, and J. Thomas for an observation. We are also grateful to an Associate Editor who suggested that we consider the case of finite state time homogeneous Markov endowments which led to Theorem 2, and to a referee for comments that improved the paper. Beker acknowledges financial support from MEC project SEJ2004-02172.