The Simple Analytics of Pure Public Good Provision

Abstract

quantities appear in his utility function. Also, the presence of the externality effectively implies a third commodity, the quantity of which influences his optimal choice and/or the maximum attainable utility in any situation. His utility function, therefore, must have three arguments. While this presents no problem for the rigorous analysis of such problems, it does complicate attempts to provide intuitive and geometric expositions of their essential features. The pure public goods model has inspired a number of devices, each useful in illustrating some particular aspect. Bowen (1943) and Samuelson (1955) noted that one could draw a demand, or marginal rate of substitution, curve for each individual, and that their vertical summation was the relevant object to equate with the marginal rate of transformation between public and private goods at a Pareto optimum. This simple construct assumed either a given distribution or a special class of preferences so as to guarantee that the optimal public goods provision was independent of distribution. Likewise, Johansen (1963) produced a graphical treatment of Lindahl equilibrium which again is generally sensitive to income redistribution. An alternative diagram has been used by McGuire and Aaron (1969) to analyse efficient public goods provision when the range of available policy instruments-specifically, the ability to make lump-sum income transfers and to choose individuals' cost shares-is limited. Pauly (1970) depicted inefficient equilibrium allocations in a similar diagram; however, Pauly's diagram does not permit an easy or direct comparison between equilibria and optima. This applies also to the treatment by Atkinson and Stiglitz (1980, p. 507), who used a two-part diagram. Other modern expositions, such as Boadway (1979), avoided diagrams at this point in their discussion. The present paper provides a thorough discussion of a relatively simple diagram which allows one to depict both the orthodox Nash equilibrium and the set of Pareto-optimal allocations in an economy with a pure public good. This diagram also allows for a simple representation of Lindahl equilibrium in such an economy. In the simple two person example, our diagram is analogous to the Edgeworth box treatment of a private goods exchange economy, and in this special case it can accommodate differences in preferences or endowments without complication. For an economy of more than two people, the diagram may still be used if the assumption of identical tastes and endowments across consumers is made. In a private goods economy with