Monday, April 27, 2015

The efficient size of a public good and why voting doesn't always work

Many people, particularly in the US, are under the impression that voting leads to the proper outcome in so far as it is the outcome that is preferred by the majority. But voting is not a panacea, and in the case of the provision of public goods it often leads to non-optimal outcomes. In this post I provide a simple example illustrating this point.

In the quest for economic efficiency neo-classical economics leaves two broad areas open to government intervention: the provision of public goods and mitigating externalities. Public goods have a specific definition in economics; they are goods that are both non-rival and non-excludable. Non-rival means that one person's enjoyment of the good does not infringe on another person's enjoyment and non-excludable means that once the good is produced it is impossible, or at least prohibitively costly, to prevent anyone from using it. National defense is usually offered up as the quintessential public good.

Some other goods that are not quite public goods but better fall under the category of club goods are often still treated as public goods for the sake of simplicity. One such example is a park. Parks can be both excludable (e.g. some national parks charge an admission fee) and rivalrous (parks can become congested). Nevertheless, parks are often examined in a public goods framework and that is what I will do here.

In the chart below I have listed three people and their willingness to pay per acre at various park sizes. Sam, Kate, and Clark all live in the same community and thus there will only be one park built, so they need to collectively decide how large to make it.


Each person has their own downward sloping demand curve that represents their willingness to pay (WTP) per acre at different park sizes. For example, Sam is willing to pay $25 per acre for a 10 acre park but only $10 per acre for a 75 acre park. The last column is society's demand (WTP) for the various park sizes. That column is the sum of each individual's willingness to pay at each park size e.g. at 10 acres $25 (Sam) + $30 (Clark) + $105 (Kate) = $160. For this example I am going to set the marginal cost of providing an acre of park land at $75. Below is a graph showing the individual demand curves, society's demand curve, and the marginal cost curve (click to enlarge).


From society's standpoint the optimal size of the park is 40 acres because that is where society's demand curve intersects the marginal cost curve of providing an acre of park land (the blue line intersects the yellow line).

At 40 acres, Sam is willing to pay $15 per acre, Clark $20, and Kate $40. But in real life people are rarely charged their willingness to pay since it is hard for the government to know exactly what that amount is. So let's suppose that the government simply divides the $75 by 3 so that each person is charged $25 per acre. At a price of $25 per acre Sam really wants 10 acres, Clark wants 25 acres, and Kate wants about 80 acres (this is where $25 intersects each of their individual demand curves).

Suppose the three of them show up to the ballot box and the question reads, "For a $25 per acre fee, how large of a park would you like?" followed by a choice of 10 acres and 25 acres. How will they vote? In this case, Sam will vote for 10 acres, Clark for 25, and Kate will also vote for 25 even though she would really like 80. So 25 acres will win.

What about a choice between 25 acres and 40 acres? Now Sam will vote for 25 acres since it is the amount closest to 10, Clark will again vote for 25, and Kate will vote for 40. So again 25 acres will win, even though from society's standpoint 40 is the optimal amount. The reason 25 wins though is because the government is not charging each person their WTP, but instead charging them all the same amount. Even if the government offered to build a very large park, say 80 acres, 25 acres would still win when paired against 80 since only Kate would vote for the 80 acre park.

Only if the choice is between 40 acres and some larger amount over 40 acres will 40 acres win. For example, if the choice is between 40 acres and 60 acres, Sam and Clark will both vote for 40 acres if the are charged $25 per acre while Kate will vote for 60 acres. In this case 40 wins. (One thing to note; this example is assuming that everyone votes on each pair of choices. In real life people can abstain from voting which would change some of the outcomes depending on who abstains. I am ignoring this complication here.)

Thus in order to get the efficient park size when charging the same amount to everyone the government will have to be strategic about the choices it offers. Providing the optimal size of a public good is difficult when the government does not know the preferences of its constituents.

So the next time you vote on a government provided good and you are given a finite amount of choices, remember that the chances of getting the optimal amount are low. Only if the government has some idea of the preferences of their constituents are they likely to provide an appropriate set of choices such that the efficient amount will be provided.

For another example about how voting does not always lead to the best outcome watch this video about Condorcet's paradox and how to rig a majority vote.

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