Is the public sector too large in an economy with club goods? A case when consumers differ in both tastes and incomes

In an archetypal economy with a single private good and a single shared good, the latter represents the public sector. With the shared good a club good, we ask if second-best (SB) provision of it is too small, as usually claimed for pure public goods. When consumers differ only in exogenous incomes, if the club good is a superior good in a single-club economy, overprovision in the SB occurs if club good demand is convex in income. We show this can extend to an economy where consumers differ in both tastes and incomes, depending on the covariances between consumers' incomes and their relative strength of preference for the club and private goods, and the covariances between incomes and taste parameters.