The provision of a public good under Cournot behavior: Stability conditions

A public goods group in which the public good is voluntarily provided by the members of that group is likely to be concerned with both the amount of the good provided and the dynamic stability of its supply. Unless the behavioral model possesses a stable equilibrium there is no guarantee that the aggregate provision will eventually stabilize at the equilibrium value. With an unstable equilibrium the group may experience continual swings in the public goods supply. In order to avoid a perpetually changing public good supply, the public good group may have an incentive to create the conditions for a stable equilibrium. In every community discussed in this paper an increase in group size beyond some maximum will produce an unstable equilibrium. A public good community may, therefore, place a limit on membership that ensures a stable equilibrium. Without knowing the properties of a stable equilibrium, it may be impossible to understand why public good groups might place restrictions on entry. For example, McGuire has shown that in a homogeneous community with a normal public good, aggregate provision will be positively related to group size. If more of the public good creates net benefits for the community, there is seemingly no reason why the group should limit entry. The present analysis provides a rational reason for exclusive normal public good clubs with limited membership. If restrictions do not exist group size might increase and produce an unstable equilibrium, a situation in which the community could not depend upon a given supply of the public good. The type of individual that the public good community might want to attract is also partly explained by the stability conditions. From McGuire's analysis it can be seen that entrants with a small absolute value for γ in their reactions functions are likely to increase the aggregate provision of the public good. In this analysis it was shown that these were precisely the same individuals who are likely to produce a stable equilibrium in the aggregate supply of the public good. Individuals with a large reaction coefficient overreact to changes in aggregate supply by the rest of the community and produce an unstable equilibrium. Such individuals may prove unattractive to the public good group for both of the above reasons; they may produce either a decrease in the aggregate provision or an unstable equilibrium. In all of the stability conditions there was a trade-off between group size and absolute value of the reaction coefficient. In other words, the permissible group size for a stable equilibrium could be increased if the absolute value of the γ i of individual members was sufficiently restricted. The acceptable range of values of γ i for a stable equilibrium becomes narrower as group size increases. This means that if larger groups wish to ensure a stable equilibrium in supply they must either spend more time and effort in the search and selection of potential members, or have some process of natural selection that restricts certain types of individuals from the public good group. Martin McGuire's analysis examined the relationship between the voluntary provision of a public good and group size. In this paper, there has been an attempt to show how stability conditions can provide further insight into the group behavior discussed by McGuire. The stability conditions contained in this paper depend on a discrete time adjustment model. If the adjustment process is changed the stability conditions will be altered. It is believed, however, that the adjustment process assumed in this paper adequately reflects the assumption of Cournot behavior; i.e., each individual adjusts his provision to the amount provided by others in the previous period. Given this adjustment process, this paper has shown the circumstances under which consumer tastes and group size affect the stability of an equilibrium position. Copyright Martinus Nijhoff Publishers bv 1980