Full and constrained Pareto efficiency with incomplete financial markets

We study the efficiency properties of equilibria in general equilibrium economies with incomplete financial markets. We focus the analysis on economies with a finitely large number of agents and initial endowments close to a Pareto optimal one. Consider an economy with a Pareto optimal equilibrium allocation and a sufficiently small, but nonzero, amount of trade. If the matrix of the derivatives of the indirect utility functions with respect to prices has maximal rank (equal to the number of non-numeraire commodities), then locally all the economies have a unique, constrained Pareto-efficient equilibrium. To the contrary, pick a Pareto optimal initial endowment. Then, locally, there are open sets of economies with a unique, constrained Pareto-efficient equilibrium and other open sets of economies with a unique constrained Pareto-inefficient equilibrium. The key step of the analysis is based on the observation that, in some small open neighborhood of an economy with a Pareto optimal initial endowment, we can partially characterize constrained Pareto optimal equilibria as the optimal solution of a well-defined, strictly concave optimization problem.