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When to provide a public good

October 23, 2011

In contrast to a private good, where the aggregate demand for the good is found by horizontally summing the demand curves of all the individuals, the aggregate demand for a public good is found by vertically summing the demand curves of all the individuals. Sometimes we are faced with the question of whether it is efficient to provide a public good. This depends on the cost of providing the public good, and the vertical sum of the demand for that public good.

We can think of this in terms of reservation prices. Whilst the non-rivalrous nature of a public good means that all individuals consume the same quantity, people can value the good differently at the margin. When the margin is the choice of whether to provide or not provide the good, the extent to which the individual values the good is that individual’s reservation price.

It is efficient to provide a public good when the sum of individuals’ reservation prices is greater than the cost of the good.

Consider the example of three students (A, B and C) who are flatsharing and are deciding whether or not to get a TV. Say it would cost them £150 to get a TV so they would split it three ways and each pay £50 if they got one. They are going to put the TV in the downstairs living room where it becomes a non-excludable public good, any of them can watch it and when it is on, the fact that one is watching it doesn’t ‘use it up’, the other two can be watching it at the same time, so it is non-rivalrous.

Suppose A places a value on having a TV in the house at £100. He is quite keen to get one, but not keen enough to fork out for it himself. B is considerably less keen, he values having a TV in the house at £40. C is fairly ambivalent about having one, he values it at £20. These are effectively their reservation prices.

If they are going to split the cost three ways and pay £50 each, then whilst A is keen to go ahead and get the TV, B and C aren’t, so if it comes down to a majority vote then A will be outvoted and the public good (the TV) won’t be provided, unless they can find a cheaper TV. A wants a TV but only values it at £100, he won’t pay the full amount himself.

However if you sum up the three students’ reservation prices you get £100 + £40 + £20 = £160, which is greater than the cost of the TV. So it be socially efficient to provide the TV. If everyone’s true preferences were revealed then A should pay £100 towards it, B should pay £40 towards it and C should pay £20 towards it and they will then have to decide what to do with the £10 change! But A would probably feel short-changed in that situation, not many of us would actually be willing to pay more than twice as much as the next person for something that they enjoy the same benefit from, even if we wanted it more. This is the problem of revealing preferences – individuals will have a tendency to understate the true amount they value the good and hope that other people provide it (ie they want to free ride off it). If everyone free rides, nobody provides the public good privately.

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