This is “Provision With Taxation”, section 8.2 from the book Beginning Economic Analysis (v. 1.0).
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Faced with the fact that voluntary contributions produce an inadequate park, the neighborhood turns to taxes. Many neighborhood associations or condominium associations have taxing authority and can compel individuals to contribute. One solution is to require each resident to contribute the amount 1, resulting in a park that is optimally sized at n, as clearly shown in the example from the previous section. Generally it is possible to provide the correct size of the public good using taxes to fund it. However, this is challenging in practice, as we illustrate in this slight modification of the previous example.
Let individuals have different strengths of preferences, so that individual i values the public good of size S at an amount that is expressed in dollars. (It is useful to assume that all people have different v values to simplify arguments.) The optimal size of the park for the neighborhood is where is the average value. Again, taxes can be assessed to pay for an optimally sized park, but some people (those with small v values) will view that as a bad deal, while others (with large v) will view it as a good deal. What will the neighborhood choose to do?
If there are an odd number of voters in the neighborhood, we predict that the park size will appeal most to the median voterThe voter whose preferences fall in the middle of the range..The voting model employed here is that there is a status quo, which is a planned size of S. Anyone can propose to change the size of S, and the neighborhood then votes yes or no. If an S exists such that no replacement gets a majority vote, that S is an equilibrium under majority voting. This is the voter whose preferences fall in the middle of the range. With equal taxes, an individual obtains If there are an odd number of people, n can be written as 2k + 1. The median voter is the person for whom there are k values vi larger than hers and k values smaller than hers. Consider increasing S. If the median voter likes it, then so do all the people with higher v’s, and the proposition to increase S passes. Similarly, a proposal to decrease S will get a majority if the median voter likes it. If the median voter likes reducing S, all the individuals with smaller vi will vote for it as well. Thus, we can see the preferences of the median voter are maximized by the vote, and simple calculus shows that this entails
Unfortunately, voting does not result in an efficient outcome generally and only does so when the average value equals the median value. On the other hand, voting generally performs much better than voluntary contributions. The park size can either be larger or smaller under median voting than is efficient.The general principle here is that the median voting will do better when the distribution of values is such that the average of n values exceeds the median, which in turn exceeds the maximum divided by n. This is true for most empirically relevant distributions.