A smooth homotopy method for incomplete markets

In the general equilibrium with incomplete asset markets (GEI) model, the excess demand functions are typically not continuous at the prices for which the assets have redundant returns. The reason is that, at these prices, the return matrix drops rank and households' budget sets collapse suddenly. This discontinuity results in a serious problem for the existence and computation of general equilibrium. In this paper, we show that this problem can be resolved with a new return matrix, which has constant rank. As a function of the price vector, the continuity of this new return matrix is ensured on a subset of the price space. This enables us to handle incomplete markets using a standard homotopy path-following argument by restricting the price vector to such a subset. The proposed approach naturally provides a constructive proof for the generic existence of general equilibrium. A homotopy method can then be applied to compute equilibria in the GEI model. Numerical experiments are presented to illustrate its efficiency.

Automated summary

The problem of discontinuity in excess demand functions in the general equilibrium with incomplete asset markets model can be resolved by introducing a new return matrix with constant rank, ensuring continuity within a subset of the price space. By restricting the price vector to this subset, standard homotopy path-following arguments can be employed to handle incomplete markets and provide a constructive proof for the generic existence of general equilibrium. A homotopy method can then be used for computing equilibria in the GEI model, with numerical experiments demonstrating its efficiency.