Chapter 37 Public Goods Key Concept: Free Riding .

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

Chapter 37 Public Goods • We argue that for certain kinds of externalities, it could be possible to eliminate inefficiency. • However, when there are many economic agents involved, private bargaining may be difficult.

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.

We now turn to a particularly troublesome kind of externality, the case of public goods. • This is a situation in which everyone must consume the same amount of the good.

For instance, streets and sidewalks are determined by local municipalities. • 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.

When to provide a public good? • We start with a simple example. • Two roommates are considering whether to buy a TV. Either they will buy a TV or they won’t. It is a 0-1 decision.

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. • So each roommate faces a budget constraint of xi + gi = wi where wi is his wealth.

Suppose the TV costs c dollars. • To buy a TV, it must be g1 + g2 ≥c. • Would it be efficient to buy a TV? • Each roommate is willing to spend ri on TV such that (wi- ri, 1) i (wi, 0).

Hence as long as r1 + r2 ≥ c, then purchasing the TV is Pareto efficient. • This suggests that the marginal willingness to pay for the TV should be the sum of each agent’s marginal willingness to pay.

Let us generalize a bit. • 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. • Each roommate faces a budget constraint of xi + gi = wi where wi is his wealth.

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).

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

This sounds intuitive. • However, 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. • 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 i is pivotal. • Let ni=vi-ci.

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

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

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

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

∑j≠i n’j≥0. ni<0. Case 2 where ∑j≠i n’j+ni<0. • Then announcing the truth ni would make i pivotal. So his payoff will be 0-∑j≠i n’j. • Announcing any n’i such that ∑j≠in’j+n’i<0 would imply the same payoff. • Announcing any n’i such that ∑j≠in’j+n’i≥0 would not make i pivotal. So his payoff is ni<-∑j≠i n’j. • 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.

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 may be susceptible to collusion. • For instance, A and B could collude to announce negative a million each. • Then the public good would not be provided. Neither is pivotal. Hence no Clarke tax is charged.

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.

Despite of that, it is a clever mechanism. • It also reminds us the second price auction.

Let us look at the case with two bidders vi and vj and bids bi and bj. • The mechanism will give the good to the bidder who bids higher. Since it is only on awarding the good, set ci = cj = 0.

When i gets the good, he is pivotal because without him, j would get the good and receive bj. This means his Clarke tax is bj. Hence i’s utility is therefore vi – bj if i gets the good. • Now, if vi > bj, then i would like to get the good. How can he achieve this? He can simply bid bi = vi > bj.

On the other hand, if vi < bj, then i would not like to get the good. Because if he gets the good, as before, he is charged the Clark tax bj. And his utility is vi – bj<0. How can he avoid this? He can simply bid bi = vi < bj. • Honesty is the best policy.

When vi>bj, he wants bi> bj. • When vi<bj, he wants bi< bj.

We now take a more general view on the Groves-Clarke mechanism. • Suppose the public good choice could be either 0 or 1 as before, or could be more than just 0 or 1. • Let x present the possible choices.

Using the notations defined earlier, each agent reports ni(x)=vi(x)-ci(x) to the center. Since they could lie, call the reported net value n’i(x). • The center chooses x* that max ∑j n’j(x). • On top of ni(x*), each agent i receives a payment of ∑j≠i n’j(x*)-maxz∑j≠i n’j(z).

Chapter 37 Public Goods Key Concept: Free Riding .