Existence and Efficiency Properties of an Approximate Equilibrium when Asset Markets are Incomplete : A Welfare Approach

We show how the general equilibrium model with incomplete asset markets can be represented by a welfare program and we use this program to prove several well-known results. The approach is general in the sense that it uses a nonlinear specification for returns on assets and does not require differentiability assumptions on utility functions. We show that the possibility of non-existence of equilibrium is linked to violation of a constraint qualification conditIOn in the welfare program, and that it cannot arise if this condition holds, but satisfaction of this condition may require (arbitrarily small) transfers between consumers, in which case the original budget constraints are only satisfied approximately. We give proofs that only use standard tools of convex programming and the Kakutani fixed point theorem. The efficiency results, as well as the well-known a..<;set pricing rule and the non-arbitrage condition are obtained as straightforward properties of the welfare program evaluated in the fixed point. Since the proofs are constructive, it is possible to compute the solution by standard fixed point algorithms