Irreducibility, taxes and the existence of equilibrium

The seminal result on the existence of competitive equilibrium in Arrow and Debreu (1954) contains, as one of its conditions, the requirement that each consumer has an endowment which is in the interior of his or her consumption set. The authors assert that such a condition, or something like it, is necessary for the existence of competitive equilibrium. They also remark that it is desirable to find a more reasonable condition than that of interior of endowments, particularly in the context of establishing the existence of competitive equilibrium. Irreducibility provides an interesting alternative to the Arrow-Debreu interior endowment condition. Florig (2001) has shown that one form of irreducibility namely, Bergstrom-Florig irreducibility, is indeed necessary for existence (at least when preferences are not price dependent). However, like the interior endowments condition, irreducibility requires that certain relationships hold across the economy – relationship which don’t have any obvious theoretical motivation and which may well fail in reality. In this paper we show that a breakdown of irreducibility can be repaired by certain tax and transfer schemes, at least as far as the existence of equilibrium is concerned.