If the problem of externalities can be solved by assigning property rights to one of the two persons involved, this may not solve the problem if there are many people which are negatively affected by the externality. If, for example, the property right is assigned to the producer of the externality and if one of the affected persons pays for a reduction in the externality generating activity, the other affected persons will also enjoy the benefits, but without having to pay anything. This is an example of a public good.
More common examples of public goods are national defense, clean air, clean streets, lighthouses etc. These are all things for which the consumption are impossible, or very expensive, to limit to only the paying person. The problem is that a market solution may not work since most people have an incentive to wait for other persons to pay for the good, and anyway enjoy the benefits. It is possible, and very likely, that too little of the public good will be forthcoming in a private market solution. The response in most societies is to substitute centrally determined provisions for decentralized market provisions of such goods.
When is it efficient for a public good to be provided? Well, we can use the notion of willingness to pay that we’ve used before to answer this question. Willingness to pay depends on the utility that person i gets from consuming a quantity x of some good; if this utility can be considered in isolation from his/hers other consumption this utility can be written as, υi(x). Person I optimal quantity x∗ is determined by setting the marginal willingness to pay, υ′i(x), equal to his or hers reservation price, ri. If the good in question is a public good, and the marginal cost of acquiring the public good is c, it is clearly efficient for person i to buy this good iff,
υ′i(x) = ri ≥ c.
Now the crucial thing about public goods are that each member of the community consumes the same amount of the public good, hence if there are n person in a small village that considers putting up street lightning for example, the condition for efficiency is that,
i.e., that the sum of the marginal willingness’s to pay is at least as great as the cost of the public good. Of course, it is often the case that the public good is quite expensive so that no single individual’s willingness to pay exceeds it’s cost. There is therefore a need for some collective decisions making.
To make things simple we assume that n = 2 and that each person has a quasi-linear utility function of the type:
Ui(xi,G) = xi + υi(G),
where x is the quantity of a private good and G the quantity of the public good. Furthermore, we’ll assume that the price of the private good is equal to 1, and that each person has an initial wealth of wi. Now, let υ1(G) = υ2(G) = 1/2G1/2 , and w1 = w2 = 1. The cost of the public good is equal to c = 3/4 . Will it be efficient for the small community (consisting of person 1 and 2) to invest in the public good? To find out, we set G = 1 and substitute into the sub-utility functions υi(1), and sum:
υ1(1) + υ2(1) = 1/2 + 1/2 = 1 > 3/4,
Hence it is efficient for them to buy the public good. But note that neither one is prepared to buy if for him or herself.
If person 1 and 2 dislike each other they may fail to buy the public good, their utility in this case is equal to U1(1, 0) = U2(1, 0) = 1. If person 1 buys the good his total utility is equal to U1(1/4 ,1) = 1/4 + 1/2 = 3/4 , but person 2’s utility (who doesn’t pay) is equal to U2(1, 1) = 3/2 . These numbers can be collected in a so called game matrix,
Reading along the first row, we can find out what the best action is for person 2, given that person 1 has already bought the public good, it’s obviously not to buy. Looking at the second row, person 2’s best action is again not to buy (1 > 3/4). Considering the first column we conclude that given that person 2 has bought the good, it is best for person 1 not to buy, and the last column tells us that if person 2 doesn’t buy, person 1 won’t either. The conclusion is that we’re stuck at the lower right-hand side cell there they each get utility of 1.
From the game matrix we can see that if person 1 buys the good and get a side payment of between 1/4 and 1/2 , both persons can be made better off, for example if the side payment is 1/3 , person 1 get’s a utility of 3/4 + 1/3 = 13/12 > 1, and person 2 a utility of 2/3 + 1/2 = 7/6 > 1. Hence, both can be better off if they can agree to share the burden of public good.
In the example above we assumed that the public good could be bought only in one discrete lump, either they bought one unit or no units. However, it is often the case that public goods can be bought in various amounts. Let’ assume that there is a third person in our small village. We assume that they have the same initial wealth (wi = 10) but that their sub-utility functions now are: (1 + i) G 1/(1+i) . If the price of the public good, per person, is equal to c = 1/4 (i.e. the share equally in the cost of the public good) the optimal amount for each person is then,
Hence, there are three different preferred levels of the public good. Now if these three persons proposed these three projects at the annual village meeting as proposals A, B and C, and that they agree to share equally in the cost of the public goods, we can rank the three proposals for each person by putting in the numbers in their utility functions (left as an exercise). If you do this you’ll find that:
1 : A > B > C2 : B > C >A3 : C > B > A
In a vote between A and B, it is clear that, B wins, in vote between A and C, C wins, in a vote between B and C, B wins. Hence in this case B is unbeatable, irrespective of how the voting is arranged (note especially that the so called voting paradox does not arise).
The previous example shows a special case when preferences are single- peaked, i.e., utility of the public good net of the cost first rise and then they decline without turning up again. If the voting mechanism is majority voting it will always be the case that the median vote will be decisive, i.e. he/she will always get his/her proposal adopted.
The level of public good provision in the case of a median voter outcome is not necessarily efficient. Person 1 wants a higher level and person 2 a lower level. For an efficient outcome we should also take into account how much more, or less, they each want and ideally they should pay different taxes. For efficiency we should have that,
υ′i(G) = θic,
where θi is person i′s tax-share. Summing over all n gives:
This is our earlier stated condition for Pareto-efficiency in the provision of public goods.
Note that with private goods each person pays the same price but consume different amounts (setting υ′i(x) = p), in the case of a public good they each consume the same amount and should (ideally) pay different prices.
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