1. Chapter 36
Public Goods

2. Public Goods -- Definition
A good is purely public if it is both nonexcludable and nonrival in consumption.
Nonexcludable -- all consumers can consume the good.
Nonrival -- one consumer’s consumption doesn’t diminish another’s.

3. Public Goods -- Examples
Broadcast radio and TV programs.
National defense.
Public highways (nearly).
Clean air.
National parks.
A good climate.

4. We will need to recall from chapter 14 that:
A consumer’s reservation price for a unit of a good is the maximum he/she is willing to pay for it.
If:
Consumer’s wealth is w
Utility when not having the good is U(w, 0)
Utility of paying p and having one unit of the good is U(w-p, 1)
Then the reservation price r is defined by

5. When Should a Public Good Be Provided?
One unit of the good costs c.
Two consumers, A and B.
Individual payments for providing the public good are gA and gB.
the good will be provided only if
gA + gB  c

6. Call the utility functions UA and UB and assume that A’s income is wA and B’s is wB.
Payments are individually rational if
and
which implies that gA ≤rA and gB ≤ rB

7. And if and then it is Pareto-improving to supply the unit of good
(It is Pareto-improving only if one inequality holds with < and the other at least with ≤)
If c ≤ rA + rB it must be pareto-efficient to supply the good.

8. Free-riding Suppose c < rA and 0 < rB .
Then A would supply the good even if B made no contribution.
Then B would enjoy the good for free; be free-riding.

9. Private Provision of a Public Good?
Suppose c < rA and c < rB .
Neither A nor B will supply the good alone.
Yet, if c < rA + rB both could be better off if the good is supplied with a payment scheme where c < gA + gB ; gA < rA and gB < rB
But A and B may try to free-ride on each other, causing no good to be supplied.

10. Free-Riding - an example
Suppose A and B each have just two actions -- individually supply a public good, or not.
Cost of supply c = 100.
Payoff to A from the good = 80.
Payoff to B from the good = 65.
> 100, so it is possible to supply the good and make both better off.

11. Player B
Don’t Buy
Buy
Buy
Player A
Don’t Buy
(Don’t’ Buy, Don’t Buy) is the unique NE

12. But (Don’t buy, Don’t buy) is not efficient.
Why?
If each pays part of the cost:
For example, A contributes 60 and B contributes 40.
Payoff to A from the good = 40 > 0.
Payoff to B from the good = 25 > 0.
Any division of 100 such that gA≤80 and
gB ≤ 65 will do.

13. Player B Don’t Contribute Contribute Contribute Player A
With cost-sharing there are two NE. But even if the good is supplied, there can still be some free-riding.
Player B
Don’t Contribute
Contribute
Contribute
Player A
Don’t Contribute

14. Variable Public Good Quantities
E.g. how many broadcast TV programs, or how much land to include into a national park.
c(G) is the production cost of G units of public good.
Two individuals, A and B.
Private consumptions are xA, xB.

15. Budget allocations must satisfy
MRSA & MRSB are A & B’s marg. rates of substitution between the private and public goods.
Pareto efficiency condition for public good supply is

16. MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good.
Similarly for B. is
the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.
the total amount of private good that A and B together are willing to give up to have G increased by 1 unit

17. If , making 1unit less of the public good releases more private good than the compensation payment requires  Pareto-improvement from reduced G.
If the payment A and B are willing to make for 1 unit of public good provides more than 1 unit  Pareto-improvement from increased G.

18. Hence, necessarily, efficient public good production requires
Suppose there are n consumers; i = 1,…,n. Then efficient public good production requires

19. Free-Riding Revisited
When is free-riding individually rational?
Individuals can contribute only positively to public good supply; nobody can lower the supply level.
Individual utility-maximization may require a lower public good level.
Free-riding is rational in such cases.

20. subject to xB + gB = wB and gB≥0
Each agent decides how much to contribute based on how much everybody else is expected to contribute.
For example:
Given A that contributes gA units of public good, B’s problem is to choose xB and gB so as to maximise UB(xB, gA+ gB)
subject to xB + gB = wB and gB≥0

21. B’s budget constraint; slope = -1
B’s endowment if A pays gA for the public good gA xB wB

22. G
B’s budget constraint; slope = -1
gB = 0 (free-riding) is best for B gA
gB<0 is not possible xB

23. Demand Revelation
A scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a revelation mechanism.
For example, the Groves-Clarke taxation scheme

24. Assume: N individuals; i = 1,…,N. All have quasi-linear preferences.
vi is individual i’s true (private) valuation of the public good.
Individual i must provide ci private good units if the public good is supplied.
ni = vi - ci is net value, for i = 1,…,N.
Can be Pareto-improving to supply the public good if

25. ni = vi - ci is net value, for i = 1,…,N.
If and or and then individual j is pivotal; i.e. changes the supply decision.

26. What loss does a pivotal individual j inflict on the others?
If then is the loss.
If then is the loss.

27. The GC tax scheme:
Assign a cost ci to each individual.
Each agent states a public good net valuation, si.
Public good is supplied if otherwise not.

28. A pivotal person j who changes the outcome from supply to not supply pays a tax of
A pivotal person j who changes the outcome from not supply to supply pays a tax of

29. GC tax scheme implements efficient supply of the public good.
Note: Taxes are not paid to other individuals, but to some other agent outside the market.

30. An example: 3 persons; A, B and C.
Valuations of the public good are: 40 for A, 50 for B, 110 for C.
Cost of supplying the good is 180.
180 < so it is efficient to supply the good.
Assign c1 = 60, c2 = 60, c3 = 60.

31. B & C’s net valuations sum to (50 - 60) + (110 - 60) = 40 > 0.
A, B & C’s net valuations sum to
( ) + 40 = 20 > 0.
So A is not pivotal.
A’s true net value is 40 – 60 = -20
If sA > -20, then A makes supply of the public good, and a loss of 20 to him, more likely.

32. If B and C are truthful, then what net valuation sA should A state?
If sA > -20, then A makes supply of the public good, and a loss of 20 to him, more likely.
If sA < -20 but not enough to make A pivotal his loss is still 20.
A prevents supply by becoming pivotal, only if sA + ( ) + ( ) < 0; To be pivotal A must state sA < -40.

33. Demand Revelation Then A suffers a GC tax of -10 + 50 = 40,
A’s net payoff is = -60 < -20.
A can do no better than state the truth; sA = -20.

34. Use the same method to show that
Exercise:
Use the same method to show that
B is not pivotal
B can do no better than state the truth;
sB = -10.
C is pivotal
C can do no better than state the truth;
sC = 50.