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Sufficient Conditions for Inessentiality

Mark Stegeman

Abstract

Consider a sequence economy in which small agents trade event-contingent claims to a single physical good. The agents have uncommon priors, state-contingent utility functions, and asymmetric information in every period. The claims traded vary from period to period. An equilibrium is inessential if, at the equilibrium prices, agents have alternate optimal plans that clear opening markets and require no trades in later markets. Theorem 1 shows that inessentiality obtains if the claims traded in opening markets span the claims traded in later markets, each agent can insure herself against improvements in her private information, and agents' initial information satisfies a weak symmetry condition. Theorem 2 shows that inessentiality obtains if opening claims satisfy a stronger, backward-looking spanning condition. Theorem 3 shows that inessentiality obtains if opening claims span payoff-relevant events, opening and later markets satisfy a statistical relationship, and agents are risk averse. None of the theorems requires ex ante Pareto optimality or the absence of arbitrage opportunities. The theorems shed light on previous questions concerning the conditions permitting speculation and the role of price-contingent trading.