Stochastic Equilibria: Existence, Spanning Number, and the "No Expected Gain from Trade" Hypothesis

Stochastic equilibria under uncertainty with continuous-time security trading and consumption are demonstrated in a general setting. The equilibrium shown has the property of no expected financial gain from trading securities. The relevence of the “no expected gain from trade” hypothesis in multi-good economies is called into question by the following demonstrated fact. For any alternative expectations a corresponding equilibrium exists, one with the original agents, original equilibrium allocations, and no expected gains from trade under the new probability assessments. The spanning number of the economy is defined as the fewest number of security markets required to sustain a complete markets equilibrium (in a dynamic sense made precise in the paper). The spanning number is linked directly to agent primitives, in particular the manner in which new information resolves uncertainty over time. The spanning number is shown to be invariant under bounded changes in expectations. Several examples are given in which the spanning number is finite even though the number of potential states of the world is infinite.