Public goods, social alternatives, and the Lindahl-VCG relationship

Lindahl prices, set by a fictitious auctioneer with full knowledge of values and costs, are a generalization of Walrasian prices. By making the efficient allocation utility- and profit-maximizing for all players, they induce an efficient outcome in a decentralized way even in the presence of public goods. We study a collective choice model with quasilinear utility, which encompasses the allocation of public and private goods as special cases. We show that each agent's most favorable Lindahl payment (the smallest Lindahl price for the efficient alternative) is equal to his VCG transfer while the firm's VCG transfer is equal to its most favorable Lindahl payment (the largest sum of Lindahl prices for the efficient alternative). Thus, the VCG mechanism incurs a deficit if and only if the set of vectors of the agents' Lindahl payments is multi-valued. Unlike Walrasian prices, Lindahl prices are not restricted to be anonymous or additive. This is the reason why, when considering the allocation of private goods, the agents' smallest Walrasian payments are at least as large as their most favorable Lindahl payments, and thus their VCG transfers. It is also why Lindahl prices always exist while Walrasian prices may not.