Private Information in the BBV Model of Public Goods

Abstract

To analyze the private provision of a public good in the presence of private information, we explore the connections between two frameworks: the binary public good model with threshold uncertainty and the standard continuous model a la Bergstrom et al. Linearity of best responses in others' contributions is key to matching the two frameworks. We identify all utility functions that display this linearity, and we provide conditions ensuring that the minimal properties that Bergstrom et al. require for utilities are satisfied. Using techniques developed in the threshold uncertainty framework, we show existence and uniqueness of the Bayes-Nash equilibrium—thus generalizing existing results—and we analyze its comparative statics properties. In particular, under the reasonable assumption that agents' income is stochastic and private information, we complement the full-information crowding-out and redistribution results of Bergstrom et al. If the government taxes agents' income proportionally and redistributes (expected) revenues lump sum, equilibrium public good provision can increase or decrease, even if the set of contributors is unchanged. Similarly, we show that crowding-out can be one-for-one, less than one-for-one, or more than one-for-one. Finally, we extend our results to a multidimensional framework in which agents' unit costs of contributions are also private information.