This is “Local Public Goods”, section 8.3 from the book Beginning Economic Analysis (v. 1.0).
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The example in the previous section showed the challenges to a neighborhood’s provision of public goods created by differences in the preferences. Voting does not generally lead to the efficient provision of the public good and does so only rarely when all individuals have the same preferences.
A different solution was proposed by TieboutCharles Tiebout, 1919–1962. His surname is pronounced “tee-boo.” in 1956, which works only when the public goods are local. People living nearby may or may not be excludable, but people living farther away can be excluded. Such goods that are produced and consumed in a limited geographical area are local public goodsGoods that are produced and consumed in a limited geographical area.. Schools are local—more distant people can readily be excluded. With parks it is more difficult to exclude people from using the good; nonetheless, they are still local public goods because few people will drive 30 miles to use a park.
Suppose that there are a variety of neighborhoods, some with high taxes, better schools, big parks, beautifully maintained trees on the streets, frequent garbage pickup, a first-rate fire department, extensive police protection, and spectacular fireworks displays, and others with lower taxes and more modest provision of public goods. People will move to the neighborhood that fits their preferences. As a result, neighborhoods will evolve with inhabitants that have similar preferences for public goods. Similarity among neighbors makes voting more efficient, in turn. Consequently, the ability of people to choose their neighborhoods to suit their preferences over taxes and public goods will make the neighborhood provision of public goods more efficient. The “Tiebout theory” shows that local public goods tend to be efficiently provided. In addition, even private goods such as garbage collection and schools can be efficiently publicly provided when they are local goods, and there are enough distinct localities to offer a broad range of services.
Consider a babysitting cooperative, where parents rotate supervision of the children of several families. Suppose that, if the sitting service is available with frequency Y, a person’s i value is viY and the costs of contribution y is ½ ny^{2}, where y is the sum of the individual contributions and n is the number of families. Rank v_{1} ≥ v_{2} ≥ … ≥ v_{n}.
What is the size of the service under voluntary contributions?
(Hint: Let yi be the contribution of family i. Identify the payoff of family j as ${v}_{j}\left({y}_{j}+{\displaystyle {\sum}_{i\ne j}{y}_{i}}\right)-\text{\xbd}n{\left({y}_{j}\right)}^{2}\text{.}$
What value of yj maximizes this expression?)
Let $\mu =\frac{1}{n}{\displaystyle \sum _{j=1}^{n}{v}_{j}}$ and ${\sigma}^{2}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}({v}_{j}}-\mu {)}^{2}\text{.}$ Conclude that, under voluntary contributions, the total value generated by the cooperative is $\frac{n}{2}\left({\mu}^{2}-{\sigma}^{2}\right)\text{\hspace{0.17em}}.$
(Hint: It helps to know that ${\sigma}^{2}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}({v}_{j}}-\mu {)}^{2}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}{v}_{j}^{2}}-\frac{2}{n}{\displaystyle \sum _{j=1}^{n}\mu {v}_{j}}+\frac{1}{n}{\displaystyle \sum _{j=1}^{n}{\mu}^{2}}=\frac{1}{n}{\displaystyle \sum _{j=1}^{n}{v}_{j}^{2}}-{\mu}^{2}.$ )