Existence of General Equilibrium in Multi-Regional Economies with Public Goods

Abstract

IN THIS PAPER WE DEMONSTRATE the existence of a general equilibrium in an economy in which the consumers are partitioned into a number of governmental jurisdictions, which provide public goods and collect taxes. The defining nature of a public good is that everyone in the jurisdiction which provides it consumes the total produced. The equilibrium concept used in this essay is due to Ellickson [3], who generalizes a definition due to Foley [4]. The essential features of the equilibrium are that producers maximize profits, consumers maximize utility subject to their after-tax budget constraint, every jurisdiction has a balanced budget, and for no jurisdiction is there an alternative public sector proposal with taxes to pay for it which is unanimously preferred by its members. We shall be dealing with the case where taxes are proportional to wealth. Ellickson offers no existence proof for such a case. Foley provides an existence proof for the special case of one jurisdiction. The central focus of this paper is the presence of regionality (i.e., the existence of more than one region). We provide an existence proof for the multi-jurisdiction case which is very different from Foley's proof for the single jurisdiction case. The techniques used in the proof would appear to have wide applicability to economies involving externalities. In Section 2 we introduce some notation and assumptions. Section 3 discusses the equilibrium concept, Section 4 gives a heuristic discussion of the strategy behind the existence proof, and Section 5 contains the formal existence proof.