Online Appendix: Omitted Proofs for Results in “optimal Provision of Multiple Excludable Public Goods” Fang and Norman (2010)

There are M excludable and indivisible public goods, labeled by j ∈ J = {1, ...,M} and n agents, indexed by i ∈ I = {1, ..., n}. The cost of providing good j, denoted Cj (n), is independent of which of the other goods are provided. Notice that Cj (n) depends on n, the number of agents in the economy, and not on the number of users. This assumption captures the fully non-rival nature of the public goods. Indexing the cost by n also enables us to analyze large economies without making the public goods a “free lunch” in the limit. We therefore assume that limn→∞C j (n) /n = cj > 0.1 Agent i is described by a valuation for each good j ∈ J . Her type is given by a vector i = ( i , ..., M i ) ∈ Θ ⊂ RM , and her preferences are represented by the utility function, ∑