Stochastic Equilibria with Incomplete Financial Markets

We demonstrate the existence of equilibria with incomplete financial markets for stochastic economies whose information structure is given by an event tree. The well known examples of non-existence of equilibria are precluded by restricting attention to purely financial securities paying in units of account, rather than “real” securities, claiming contingent commodities. Financial markets may be incomplete: some consumption streams may be impossible to obtain by any trading strategy. Securities may be individually precluded from trade at arbitrary sets of nodes (states and dates). Sufficient conditions for the existence of stochastic equilibria are: continuous, convex, strictly monotonic preferences; and wealth accessibility: the existence of a trading strategy for each agent that, in conjunction with endowment market values, leaves strictly positive wealth at each node. A sufficient (but not necessary) condition for wealth accessibility is positive non-zero endowments for each agent at each node, and strictly positive aggregate endowments. A corollary states that any regime of security prices precluding arbitrage can be embedded in an equilibrium. That is, with fixed security prices, spot goods prices can be adjusted to clear both spot and security markets.