On strong independence of allocative efficiency from distribution in the theory of public goods

We extend Bergstrom and Cornes (1983) to show that for strong independence of efficient allocations from distribution in a public goods economy, the utility functions of all consumers must identically be of the form: A(Y)Xi, where Y and Xi are respectively the quantities of public good and private good for consumer i, and A(⋅) is some arbitrary function. This implies that for an economy with heterogeneous consumer preferences, it is impossible to ensure that any redistribution of private goods will remain efficient, especially for boundary Pareto optima.