The private provision of public good in the case of satiation points: The case of a quasi-linear economy

We consider a quasi-linear economy with satiation points for the public good which can be different from an individual to another. We prove that if the range of the support of the distribution of satiation points is greater than the average of satiation points, there always exist wealth distributions for which the associated Nash equilibrium of voluntary contributions gives an overproduction of the public good relatively to the Pareto optimal quantity. Surprisingly we prove under weak assumptions that the probability of an oversupply is strictly positive but remains smaller that the probability of an undersupply. This result holds for all distributions of satiation points such that the median is smaller or equal to the arithmetic mean and for any distributions of initial wealth. FUrthermore, if the distribution of satiation points is a truncated normal or lognormal distribution we obtain the following result: the probability of an overproduction of the Nash quantity respectively to the Pareto optimal quantity is all the stronger since the satiation quantities of the public good are small relative to the resources of the economy. A stochastic dominance argument is used in the proof.