Question. A community consisting of two house¬holds must decide whether to build a swimming pool or not. Building a pool costs C = 8. Each household's valuation for the pool is either vi = 1 (with probability 1/3) or = 10 (with probability 2/3) and independent of the other household's valuation. These facts are commonly known; however, each household alone knows its actual valuation. The pool is a non-rivalrous good. A household that uses the pool receives utility equal to its valuation minus any payments it needs to make. A household that does not use the pool receives utility equal to the negative of any payments it needs to make.

(a) When is it efficient to build the pool? What is the expected social welfare the pool can generate for the community?

Suppose the conununity is governed by a neighborhood association that has the right to levy (possibly differentiated) taxes on households for the public amenities it provides. The association provides the pool as a public good. To decide whether to construct the pool the association surveys the households, asking them how much they would value a swimming pool in the neighborhood. Note that, since valuations are privately known, households may not report them truthfully. The association's goals are to provide the pool efficiently, fund it fully (if built), and never tax any household more than its valuation.

(b) Show that it is impossible to satisfy the association's goals simultaneously if house-holds play (Bayesian) Nash equilibrium when answering the survey.

(c) Consider the following alternative scheme: If both neighbors report the same valua-tion, the pool is provided efficiently; if one neighbor reports 1 and the other reports 10, the pool is provided with probability p. Assume the pool is stall fully funded and taxes are individually rational. Find the largest p for which individuals are induced to reveal their valuations titthfully.

Now suppose that the association operates the swimming pool as a club. That is, instead of taxing the householdiA, it sells access to the pool through (possibly differenti¬ated) membership fees. Every households is free to buy a membership or not. Before deciding whether to build the pool and how to structure its fees, the association surveys the individuals as in (b)- (c). The &Rsociation'a goals are to operate the club efficiently and fund k fully.

(d) Show that it is impossible to satisfy the association's goals simultaneously if house-holds play (Bayesian) Nash equilibrium when answering the survey.

(e) Consider the following alternative scheme; Households are surveyed and the pool built if and only if it is efficient to do so. Club membership is then determined as follows: Households who reported 10 are invited to join, and households who reported I are invited to join with probability r. Invited households decide whether to join, and a fee structure is put in place that fully funds the pool. Compute the largest r such that households reveal their valuations truthfully.

We now compare the two provision/financing schemes: First-best as in (a), taxation as in (c), and club membership as in (e).

(f) For (c) and (e), compute the expected social welfare generated by the swimming pool. You should find that both are lower than the welfare in (a), but welfare is higher in (e) than in (c). Explain this ordering.