A Constructive Proof of the Existence of Collateral Equilibrium for a Two-Period Exchange Economy Based on a Smooth Interior-Point Path

Abstract

In the real economic world, we often suffer from the default risk—the present financial crisis could serve as an evidence. The long established idea for inhibiting the default behavior is to resort to collateral, and this idea has profound effect on the market equilibrium. To study this effect, we make an investigation of the economy wherein default risk is present, assets are collateralized and households are risk-averse. We shall prove that the economy is in possession of an equilibrium. This proof is distinct from the existing one, in two respects: First it shows that equilibrium exists for the Leontief utility function, and second it is constructive. By constructive we mean that, on the basis of this proof, we can (and we shall) develop an algorithm for computing that equilibrium. The algorithm developed is of a path-following type. It starts from an economy which consists of several sub-economies, each with a complete market and a Cobb–Douglas utility function; and it terminates at the equilibrium of the economy under consideration. The algorithm is shown by simulation to be effective.