Chapter 37 Public Goods Key Concept: Free Riding. The Groves-Clarke mechanism asks people to be responsible for their externalities.

Chapter 37 Public Goods Suppose there are two nonsmokers and one smoker. Even if nonsmokers are entitled to clean air, they first have to agree among themselves how much smoke should be allowed and what the compensation should be. It is typically a difficult issue.

National defense is another typical example National defense is another typical example. Government is often involved to determine the level of public goods. However, bear in mind the determining the level by the government is one thing, providing (constructing) the public good is another thing.

Let x1 and x2 measure the private consumption of each person. Let g1 and g2 be their contributions to a public good, say the TV. Let G measure the quality of TV they buy. Let the cost function be c(G).

The Pareto efficient allocation should satisfy: maxx1,x2,G u1(x1,G) s.t. u2(x2,G)=k and x1+x2 +c(G)= w1+w2. FOC |MRSG,x1|+|MRSG,x2|=c’(G).

FOC |MRSG,x1|+|MRSG,x2|=c’(G). Intuitively, to increase one unit of G, 1 is willing to give up |MRSG,x1| dollars and 2 |MRSG,x2|, since society has to pay c’(G) for the marginal unit, hence the sum of MRS should equal the marginal cost to be efficient.

Hence we vertically sum the MRS to derive the MRS of the society and intersect it with the MC to get an efficient amount of G.

Fig. 36.1

Two things are worth mentioning. To provide public goods, we often run into the free rider problem. Look at two roommate example. Suppose C(G)=G for simplicity.

Consumer 1’s problem is maxx1, g1 u1(x1, g1+g2) s.t. x1+g1= w1. Consumer 2’s problem is maxx2, g2 u2(x2, g1+g2) s.t. x2+g2=w2.

maxx1, g1 u1(x1, g1+g2) s.t. x1+g1= w1. maxx2, g2 u2(x2, g1+g2) s.t. x2+g2=w2. Now it is quite likely that consumer 1 is a crazy TV fan and his optimal consumption of g1 is large enough that consumer 2 will choose to free ride and set g2=0. This is a Nash equilibrium between the two roommates, however it is clearly not efficient.

Fig. 36.2

So the key problem becomes how we can elicit people’s true preferences about the public goods?

Groves-Clarke mechanism may be a partial answer to that.

Suppose we are considering whether to buy a public good. Let us suppose the public good will cost 100 dollars. Each person has a valuation vi for the public good.

So the public good should be provided if ∑ivi ≥100 So the public good should be provided if ∑ivi ≥100. Now, how can we elicit people’s vi?

We can first assign a cost ci to each individual, and the individual will have to pay if it is decided that the public good will be provided. (∑ici=100) Then have each individual state his value vi. If the sum of the values is ≥100, it will be provided. If it is <100, it will not be.

Most importantly, we calculate whether any individual j is pivotal. Let ni=vi-ci.

Let ni=vi-ci. If without j, ∑i≠jni≥0, but ∑ini<0, then j is pivotal. So he has to pay a tax of ∑i≠jni. Similarly, if without j, ∑i≠jni<0, but ∑ini≥0, then j is pivotal. So he has to pay a tax of -∑i≠jni. The idea is to make j responsible for the externalities he exerts on others.

Now why will j reveal his true preference if the mechanism is in use?

Suppose any other player i≠j announces n’i and ∑i≠jn’i≥0 Suppose any other player i≠j announces n’i and ∑i≠jn’i≥0. If his true preference implies that nj<0. There are two possible cases. Case 1 where ∑i≠jn’i+nj≥0. Case 2 where ∑i≠jn’i+nj<0.

∑i≠jn’i≥0. nj<0. Case 1 where ∑i≠jn’i+nj≥0. Then announcing the truth nj would not make j pivotal. So his payoff will be nj. Announcing any n’j such that ∑i≠jn’i+n’j≥0 would imply the same payoff. Announcing any n’j such that ∑i≠jn’i+n’j<0 would make j pivotal. So his payoff is 0-∑i≠jn’i≤nj. Hence honesty is the best policy.

∑i≠jn’i≥0. nj<0. Case 2 where ∑i≠jn’i+nj<0. Then announcing the truth nj would make j pivotal. So his payoff will be 0-∑i≠jn’i. Announcing any n’j such that ∑i≠jn’i+n’j<0 would imply the same payoff. Announcing any n’j such that ∑i≠jn’i+n’j≥0 would not make j pivotal. So his payoff is nj<-∑i≠jn’i. Hence honesty is the best policy.

You can work out any other possibilities and convince yourself that honesty is the best policy. Let us work out a numerical example.

person ci vi ni Clarke tax A 100 50 -50 B C 250 150

A has to announce something (vi) less than 0 to make the public good not being provided. In this case, he is pivotal and so his payoff will be -(-50+150)=-100. Not worth of doing it since by telling the truth, he is getting -50. C has to announce something (vi) less than 200 to make the public good not being provided. In this case, he is not pivotal and so his payoff will be 0. But by telling the truth he is getting 150-100=50.

It looks pretty good but there are potential problems.

Quasilinear preferences since we cannot have the amount that you have to pay influence your demand for the public good.

Not necessarily efficient in the private good. The Clarke tax cannot go to anyone for if so, it might affect their decisions. Yet this is less serious when there are many people.

It is not the case that everyone is better off It is not the case that everyone is better off. Consider the previous example, A and B are worse off.

If instead we have the cost allocation to be 40 40 220, then everyone is better off. But notice that we start with the situation that we don’t know what the valuations are, so it might not be possible for us to make such a wise choice of cost allocation at the first place.

Chapter 37 Public Goods Key Concept: Free Riding. The Groves-Clarke mechanism asks people to be responsible for their externalities.