The public choice literature is generally pessimistic about the efficacy of entrusting the provision of public goods to individualistic voluntary arrangements. Analysis frequently ends with a description of the free rider problem and the prisoner's dilemma. However, Buchanan (1968) has given a Pr0babilistic analysis of the free rider problem that suggests that there is more in the way of analysis that can be applied. Buchanan alludes to previous work by Buchanan (1965b) and Olson (1965) that suggests that voluntary provision of public good is more likely to be satisfactory in small groups than large groups and then presents a model of the provision of public goods through voluntary contributions. In the large group case (with extreme lumpiness) no provision of the public good is forthcoming through voluntary contributions -a result that has subsequently been given extensive analysis by McGuire (1974) and Roberts (1976). However, Buchanan points out that with small groups and/or divisibility of the public good some quantity of the public good might be forthcoming. The purpose of this note is to provide a somewhat generalized formalization of the Buchanan model. The results of this generalization do not contradict previous analysis but they do provide some special insight into the theory of clubs (Buchanan, 1965;Ng, 1973) and sufficient conditions for Pareto optimality where a public good is provided through voluntary contributions.
ยฐ To keep the model simple we will assume one private good, one public good, and n individuals. Both goods are produced by a single factor of production (time) which can be consumed directly (as leisure) or used for production (as labor). The private good is produced by a commercial sector and priced at marginal cost. The public good is produced through an arrangement whereby individuals make voluntary contributions to a fund that is used to purchase the factor of production in the open market for the