1. Chapter Thirty-Five
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 -- each consumer can consume all of the good.

3. Public Goods -- Examples
Broadcast radio and TV programs.
National defense.
Public highways.
Reductions in air pollution.
National parks.

4. Reservation Prices
A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it.
Consumer’s wealth is
Utility of not having the good is

5. Reservation Prices
A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it.
Consumer’s wealth is
Utility of not having the good is
Utility of paying p for the good is

6. Reservation Prices
A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it.
Consumer’s wealth is
Utility of not having the good is
Utility of paying p for the good is
Reservation price r is defined by

7. Reservation Prices; An Example
Consumer’s utility is
Utility of not buying a unit of good 2 is
Utility of buying one unit of good 2 at price p is

8. Reservation Prices; An Example
Reservation price r is defined by
I.e. by

9. 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.
gA + gB  c if the good is to be provided.

10. When Should a Public Good Be Provided?
Payments must be individually rational; i.e. and

11. When Should a Public Good Be Provided?
Payments must be individually rational; i.e. and
Therefore, necessarily and

12. When Should a Public Good Be Provided?
And if and then it is Pareto-improving to supply the unit of good

13. When Should a Public Good Be Provided?
And if and then it is Pareto-improving to supply the unit of good, so is sufficient for it to be efficient to supply the good.

14. Private Provision of a Public Good?
Suppose and
Then A would supply the good even if B made no contribution.
B then enjoys the good for free; free-riding.

15. Private Provision of a Public Good?
Suppose and
Then neither A nor B will supply the good alone.

16. Private Provision of a Public Good?
Suppose and
Then neither A nor B will supply the good alone.
Yet, if also, then it is Pareto-improving for the good to be supplied.

17. Private Provision of a Public Good?
Suppose and
Then neither A nor B will supply the good alone.
Yet, if also, then it is Pareto-improving for the good to be supplied.
A and B may try to free-ride on each other, causing no good to be supplied.

18. Free-Riding
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.

19. Free-Riding
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.
$80 + $65 > $100, so supplying the good is Pareto-improving.

20. Free-Riding
Player B
Don’t Buy
Buy
Buy
Player A
Don’t Buy

21. Free-Riding Player B Don’t Buy Buy Buy Player A Don’t Buy
(Don’t’ Buy, Don’t Buy) is the unique NE.

22. Free-Riding Player B Don’t Buy Buy Buy Player A Don’t Buy
But (Don’t’ Buy, Don’t Buy) is inefficient.

23. Free-Riding
Now allow A and B to make contributions to supplying the good.
E.g. A contributes $60 and B contributes $40.
Payoff to A from the good = $40 > $0.
Payoff to B from the good = $25 > $0.

24. Free-Riding Player B Don’t Contribute Contribute Contribute Player A

25. Free-Riding Player B Don’t Contribute Contribute Contribute Player A
Two NE: (Contribute, Contribute) and (Don’t Contribute, Don’t Contribute).

26. Free-Riding
So allowing contributions makes possible supply of a public good when no individual will supply the good alone.
But what contribution scheme is best?
And free-riding can persist even with contributions.

27. Variable Public Good Quantities
E.g. how many broadcast TV programs, or how much land to include into a national park.

28. 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.

29. Variable Public Good Quantities
Budget allocations must satisfy

30. Variable Public Good Quantities
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

31. Variable Public Good Quantities
Pareto efficiency condition for public good supply is
Why?

32. Variable Public Good Quantities
Pareto efficiency condition for public good supply is
Why?
The public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B.

33. Variable Public Good Quantities
Suppose
MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good.
Similarly for B.

34. Variable Public Good Quantities
is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.

35. Variable Public Good Quantities
is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.
Since , making 1 less public good unit releases more private good than the compensation payment requires  Pareto-improvement from reduced G.

36. Variable Public Good Quantities
Now suppose

37. Variable Public Good Quantities
Now suppose
is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.

38. Variable Public Good Quantities
Now suppose
is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.
This payment provides more than 1 more public good unit  Pareto-improvement from increased G.

39. Variable Public Good Quantities
Hence, necessarily, efficient public good production requires

40. Variable Public Good Quantities
Hence, necessarily, efficient public good production requires
Suppose there are n consumers; i = 1,…,n. Then efficient public good production requires

41. Efficient Public Good Supply -- the Quasilinear Preferences Case
Two consumers, A and B.

42. Efficient Public Good Supply -- the Quasilinear Preferences Case
Two consumers, A and B.
Utility-maximization requires

43. Efficient Public Good Supply -- the Quasilinear Preferences Case
Two consumers, A and B.
Utility-maximization requires
is i’s public good demand/marg. utility curve; i = A,B.

44. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUB
MUA G

45. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MUA G

46. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MC(G)
MUA G

47. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MC(G)
MUA G* G

48. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MC(G)
pG*
MUA G* G

49. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MC(G)
pG*
MUA G* G

50. Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA+MUB
MUB
MC(G)
pG*
MUA G* G
Efficient public good supply requires A & B
to state truthfully their marginal valuations.

51. Free-Riding Revisited
When is free-riding individually rational?

52. Free-Riding Revisited
When is free-riding individually rational?
Individuals can contribute only positively to public good supply; nobody can lower the supply level.

53. 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.

54. Free-Riding Revisited
Given A contributes gA units of public good, B’s problem is subject to

55. Free-Riding Revisited
B’s budget constraint; slope = -1 gA xB

56. Free-Riding Revisited
B’s budget constraint; slope = -1 gA
is not allowed xB

57. Free-Riding Revisited
B’s budget constraint; slope = -1 gA
is not allowed xB

58. Free-Riding Revisited
B’s budget constraint; slope = -1 gA
is not allowed xB

59. Free-Riding Revisited
B’s budget constraint; slope = -1
(i.e. free-riding) is best for B gA
is not allowed xB

60. Demand Revelation
A scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a revelation mechanism.
E.g. the Groves-Clarke taxation scheme.
How does it work?

61. Demand Revelation 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.

62. Demand Revelation ni = vi - ci is net value, for i = 1,…,N.
Pareto-improving to supply the public good if

63. Demand Revelation ni = vi - ci is net value, for i = 1,…,N.
Pareto-improving to supply the public good if

64. Demand Revelation
If and or and then individual j is pivotal; i.e. changes the supply decision.

65. Demand Revelation
What loss does a pivotal individual j inflict on others?

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

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

68. Demand Revelation
For efficiency, a pivotal agent must face the full cost or benefit of her action.
The GC tax scheme makes pivotal agents face the full stated costs or benefits of their actions in a way that makes these statements truthful.

69. Demand Revelation 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.

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

71. Demand Revelation
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

72. Demand Revelation
Note: Taxes are not paid to other individuals, but to some other agent outside the market.

73. Demand Revelation
Why is the GC tax scheme a revelation mechanism?

74. Demand Revelation Why is the GC tax scheme a revelation mechanism?
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.

75. Demand Revelation Why is the GC tax scheme a revelation mechanism?
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 < $40 + $50 + $110 so it is efficient to supply the good.

76. Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.

77. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
B & C’s net valuations sum to $( ) + $( ) = $40 > 0.
A, B & C’s net valuations sum to
$( ) + $40 = $20 > 0.

78. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
B & C’s net valuations sum to $( ) + $( ) = $40 > 0.
A, B & C’s net valuations sum to
$( ) + $40 = $20 > 0.
So A is not pivotal.

79. Demand Revelation
If B and C are truthful, then what net valuation sA should A state?

80. Demand Revelation
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.

81. Demand Revelation
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.
A prevents supply by becoming pivotal, requiring sA + $( ) + $( ) < 0; I.e. A must state sA < -$40.

82. Demand Revelation Then A suffers a GC tax of -$10 + $50 = $40,
A’s net payoff is - $20 - $40 = -$60 < -$20.

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

84. Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.

85. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
A & C’s net valuations sum to $( ) + $( ) = $30 > 0.
A, B & C’s net valuations sum to
$( ) + $30 = $20 > 0.

86. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
A & C’s net valuations sum to $( ) + $( ) = $30 > 0.
A, B & C’s net valuations sum to
$( ) + $30 = $20 > 0.
So B is not pivotal.

87. Demand Revelation
What net valuation sB should B state?

88. Demand Revelation What net valuation sB should B state?
If sB > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.

89. Demand Revelation What net valuation sB should B state?
If sB > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.
B prevents supply by becoming pivotal, requiring sB + $( ) + $( ) < 0; I.e. B must state sB < -$30.

90. Demand Revelation Then B suffers a GC tax of -$20 + $50 = $30,
B’s net payoff is - $10 - $30 = -$40 < -$10.
B can do no better than state the truth; sB = -$10.

91. Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.

92. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
A & B’s net valuations sum to $( ) + $( ) = -$30 < 0.
A, B & C’s net valuations sum to
$( ) - $30 = $20 > 0.

93. Demand Revelation Assign c1 = $60, c2 = $60, c3 = $60.
A & B’s net valuations sum to $( ) + $( ) = -$30 < 0.
A, B & C’s net valuations sum to
$( ) - $30 = $20 > 0.
So C is pivotal.

94. Demand Revelation
What net valuation sC should C state?

95. Demand Revelation What net valuation sC should C state?
sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0.

96. Demand Revelation What net valuation sC should C state?
sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0.
sC < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.

97. Demand Revelation What net valuation sC should C state?
sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0.
sC < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.
C can do no better than state the truth; sC = $50.

98. Demand Revelation
GC tax scheme implements efficient supply of the public good.

99. Demand Revelation
GC tax scheme implements efficient supply of the public good.
But, causes an inefficiency due to taxes removing private good from pivotal individuals.