Remarks

We now have several tools at our disposal for proving the existence of a Walrasian equilibrium of an economy. There are many ways open to do this. We will focus on two approaches. Other approaches will be described and references given at the end of this chapter. The two approaches are the excess demand approach and the abstract economy approach. The excess demand approach utilizes the Debreu-Gale-Nikaido lemma (18.1). The abstract economy approach converts the problem of finding a Walrasian equilibrium of the economy into the problem of finding the Nash equilibrium of an associated abstract economy.

The central difficulty of the excess demand approach involves proving the upper hemi-continuity of the excess demand correspondence. The maximum theorem (12.1) is used to accomplish this, but the problem that must be overcome is the failure of the budget correspondence to be lower hemi-continuous when a consumer's income is at the minimum compatible with his consumption set (cf. 11.18(e)). When this occurs, the maximum theorem can no longer be used to guarantee the upper hemi-continuity of the consumer's demand correspondence. There are two ways to deal with this problem. The first is to assume it away, by assuming each consumer has an endowment large enough to provide him with more than his minimum income for any relevant price vector. The other approach is to patch up the demand correspondence's discontinuities at places where the income reaches its minimum or less, then add some sort of interrelatedness assumption on the consumers to guarantee that in equilibrium, they will all have sufficient income.