Pareto efficient (optimal) level of provision of public good is such that cannot be altered in any way so that at least one person would be made better off without making anybody worse off. On the other hand, Pareto inefficient provision is such that there exists a way to make at least one individual better off (so called Pareto improvement) without inflicting any harm on others. Let us take a look at an example to illustrate what this means.

• Whether or Not to Buy a Shared TV.
• How Much to Provide?
• Private Provision of the Public Good and Free Riding
• Advanced Discussion of The Individual's Public Good Choice Problem in the Market

Suppose there are two roommates, Al and Brad, who are deciding to buy a common TV. The aspect of sharing makes the TV a public good, rather than a private good. It is very likely that the two roommates value the TV differently. To measure the value of TV to each of them we will use their willingness to pay for the item. The willingness to pay for Al and Brad is the maximum price each of them is willing to pay in order to obtain the TV, that is leaving them both indifferent between paying their [LINK reservation prices] or not purchasing the TV at all.

Let w_A and w_B denote Al's and Brad's initial wealth and r_A and r_B their respective reservation prices. If Al pays his reservation price r_A for the TV he will have w_A - r_A left for private consumption. If he does not purchase the TV, he will have w_A available to spend on private consumption. Being indifferent between these two options implies that u_A(w_A - r_A, 1) = u_A(w_A, 0), where 1 (0) means that he is consuming one (zero) unit(s) of the public good. Brad's reservation price can be defined in a similar way.

We are interested in two different types of allocation - one when the public good is provided, (w_A - g_A, w_B - g_B, 1), and one when is not, (w_A, w_B, 0), where g_A and g_B represent Al's and Brad's contributions towards the public good. When will the two roomamtes be better off buying a TV comparing to a situation when they spent their wealth on private consumption only?

Al and Brad should decide to purchase a TV if they are both better off having it and paying their contribution than not being able to watch TV at home. In economic terms:
u_A(w_A, 0) < u_A(w_A - g_A, 1)
u_B(w_B, 0) < u_B(w_B - g_B, 1).

Using the reservation prices r_A and r_A:
u_A(w_A, 0) = u_A(w_A - r_A, 1) < u_A(w_A - g_A, 1)
u_B(w_B, 0) = u_B(w_B - r_B, 1) < u_B(w_B - g_B, 1).

Given that higher private consumption increases utility it follows from the above inequalities that
w_A - r_A < w_A - g_A
w_B - r_B < w_B - g_B,

which implies that
r_A > g_A
r_B > g_B.

From the above condition we can conclude when it is a good idea for the two roommates to obtain the TV. If buying the TV is a Pareto improvement, then it must be that each person's contribution to the public good is less then his willingness to pay for it. It means that both Al and Brad will be able to get the TV for less than the maximum they are willing to pay for it (given its cost is smaller than the sum of their reservation prices). Then there always exist some sontribution scheme under which they will be better off having the TV than not having it.

References:

Hal R. Varian: Intermediate Economics - A Modern Approach, 6th ed., Norton 2002.

Suppose there are several levels at which the public good can be provided. What is the optimal level of provision then? The following example illustrates such case.

Suppose Amy and Bev live on a lake and are plagued by mosquitoes, but it is possible to control the mosquitoes by spraying regularly. The mosquito spray cannot be confined to the property of just one of the women: any spray that is used affects the entire lake shore equally. Suppose that Amy is willing to pay at most $28 for the first gallon of bug spray, $26 for the second, $24 for the third etc. and Bev is willing to pay at most $5 for the first gallon, $4 for the second, $3 for the third and so on. Suppose also that the spray costs $24 per gallon. How much of the bug spray should they buy?

To solve Amy's and Bev's problem we have to look at the marginal social value of each gallon of spray and compare it with the social marginal cost. The first gallon Amy and Bev value together at $28 + $5 = $33. The spray costs $24 per gallon. The girls value it more than it costs to obtain and thus they should purchase this unit. The second gallon has marginal social value of $26 + $4 = $30, third $24 + $3 = $27, fourth $22 + $2 = $24. Since Amy and Bev value all four units more than or equally to their marginal cost, they should buy exactly four units of bug spray, because that is the socially optimal level. Notice that this Pareto efficient amount of spray equates the marginal social value to the marginal social cost. Taking this argument further, Amy is willing to pay only $20 and Bev $1 for the fifth gallon of spray. The marginal social value yields $21 and at the cost of $24 it is inefficient to purchase another unit.

The bug-spray example is also discussed in more detail in Public Goods Exercises section.

How much will be sprayed if Amy and Bev each choose on the basis of just their own personal benefits from the spray? Bev will not choose to spray at all, because even when x = 0 her value of bug spray is much less than the price of it. Amy will choose x = 3 gallons of spray, which equates her own value of teh third unit to the price of the spray. Thus, too little spray is chosen. Both women would be better off if an additional gallon were sprayed and, say, Amy paid $22 of the cost and Bev paid the remaining $2.

Notice that when people reason about their own optimal level, it usually yields an underprovission of the public good from the social point of view. The inefficiency is caused by the fact that when optimizing at the private level the individual cannot often purchase the public good because the private marginal cost exceeds the private marginal value resulting in zero purchase by that person and thus not contributing towards the provision of the public good.

Another important observation is that individuals have an incentive not to contribute towards the public good because they rely on other persons' contributions they will benefit from without paying anything. Such behavior is called free riding. An illustration of free riding is not revealing the true valuation of TV by either Al or Brad in the above example and hoping that the other person would go and purchase the TV by himself anyway. In this case the free rider would not have to pay anything (or less than under some other payment scheme when both contribute based on their true valuations), but could still enjoy the good. Another example of free riding can be found in the Voluntary Contribution Mechanism subsection. More advanced discussion on private provision of public good vs. Pareto efficiency is provided in the Individual's Public Good Choice Problem in the Market subsection.