Non-existence of recursive equilibria on compact state spaces when markets are incomplete

This paper analyzes one-good exchange economies with two infinitely-lived agents and incomplete markets. It is shown that there are no recursive (Markov) equilibria for which borrowing (debt) constraints never bind if the state space of exogenous and endogenous variables is a compact subset of Rn. Moreover, for large enough (but finite) borrowing limits, no recursive equilibrium with compact state space exists. These non-existence results hold for any economy satisfying the following standard assumptions: preferences are additively separable across time and states; the one-period utility function is time- and state-independent and unbounded from below; endowments are bounded and follow a Markov chain (finite support) with stationary transition matrix; there is some idiosyncratic risk and no aggregate risk. The same non-existence results hold for any number of agents if in addition the marginal utility functions of all agents are convex. Finally, non-existence holds for any number of agents and any Markov endowment chain (with or without aggregate risk) if all agents have identical utility functions of the CRRA-type (homothetic preferences) with degree of relative risk aversion greater than or equal to one.