A New Characterization of Equilibrium in a Multi-period Finance Economy: A Computational Viewpoint

Computing equilibrium in incomplete-markets with long-lived assets is a challenging task especially because equilibrium may not exist due to the ‘bad-price’ problem. When algorithms fail to produce equilibrium outcomes, it is hard to differentiate between the existential failure and the algorithmic failure. Moreover, algorithmic success can be misleading when algorithms work out a quasi-solution for the system of equations which may fail to have equilibrium. To address the computational dilemma, the paper provides a new approach to computing equilibrium in a multi-period, single-good general equilibrium model with incomplete asset markets (single-good GEI model or stochastic finance model). The new approach is built on a notion of ‘pre-GEI equilibrium’ which always exists in the economy. When the payoff matrix has full rank, equilibrium of the stochastic finance economy (GEI equilibrium) coincides with pre-GEI equilibrium in real terms. This implies full-rank GEI equilibrium can be computed as pre-GEI equilibrium. It is shown that pre-GEI equilibrium is determined in a system of equations which can be encoded into diverse algorithms such as homotopy path-following methods. Since pre-GEI equilibrium always exists, the existential failure cannot occur and thus, computational failures imply algorithmic failures in the current computational framework.