Chapter Thirty-Six

1
Chapter Thirty-Six

• 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 consumers reservation price for a unit of a
  good is his maximum willingness-to-pay for it.
• Consumers wealth is
• Utility of not having the good is

5
Reservation Prices

• A consumers reservation price for a unit of a
  good is his maximum willingness-to-pay for it.
• Consumers wealth is
• Utility of not having the good is
• Utility of paying p for the good is

6
Reservation Prices

• A consumers reservation price for a unit of a
  good is his maximum willingness-to-pay for it.
• Consumers 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
Consumers utility is
Utility of not buying a unit of good 2 is
Utility of buying one unit of good 2 atprice 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 ifandthen it is Pareto-improving to supply
  the unit of good

13
When Should a Public Good Be Provided?

• And ifandthen 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 gt 100, so supplying the good is
  Pareto-improving.

20
Free-Riding
Player B
DontBuy
Buy
Buy
Player A
DontBuy
21
Free-Riding
Player B
DontBuy
Buy
Buy
Player A
DontBuy
(Dont Buy, Dont Buy) is the unique NE.
22
Free-Riding
Player B
DontBuy
Buy
Buy
Player A
DontBuy
But (Dont Buy, Dont 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 20 gt 0.
• Payoff to B from the good 25 gt 0.

24
Free-Riding
Player B
DontContribute
Contribute
Contribute
Player A
DontContribute
25
Free-Riding
Player B
DontContribute
Contribute
Contribute
Player A
DontContribute
Two NE (Contribute, Contribute) and (Dont
Contribute, Dont 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 Bs 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 As 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 is public good
  demand/marg. utility curve i A,B.

44
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUB
MUA
G
45
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MUB
MUA
G
46
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MC(G)
MUB
MUA
G
47
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MC(G)
MUB
MUA
G
G
48
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MC(G)
MUB
pG
MUA
G
G
49
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MC(G)
MUB
pG
MUA
G
G
50
Efficient Public Good Supply -- the Quasilinear
Preferences Case
pG
MUAMUB
MC(G)
MUB
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, Bs
  problem issubject to

55
Free-Riding Revisited
G
Bs budget constraint slope -1
gA
xB
56
Free-Riding Revisited
G
Bs budget constraint slope -1
gA
is not allowed
xB
57
Free-Riding Revisited
G
Bs budget constraint slope -1
gA
is not allowed
xB
58
Free-Riding Revisited
G
Bs budget constraint slope -1
gA
is not allowed
xB
59
Free-Riding Revisited
G
Bs 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 is 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 andor
  andthen 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 supplypays a tax of

71
Demand Revelation

• A pivotal person j who changes the outcome from
  supply to not supplypays a tax of
• A pivotal person j who changes the outcome from
  not supply to supplypays 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 are40 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 are40 for A, 50
  for B, 110 for C.
• Cost of supplying the good is 180.
• 180 lt 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 Cs net valuations sum to(50 - 60) (110
  - 60) 40 gt 0.
• A, B Cs net valuations sum to
• (40 - 60) 40 20 gt 0.

78
Demand Revelation

• Assign c1 60, c2 60, c3 60.
• B Cs net valuations sum to(50 - 60) (110
  - 60) 40 gt 0.
• A, B Cs net valuations sum to
• (40 - 60) 40 20 gt 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 gt -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 gt -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 (50 - 60) (110 - 60) lt 0I.e. A must
  state sA lt -40.

82
Demand Revelation

• Then A suffers a GC tax of -10 50 40,
• As net payoff is - 20 - 40 -60 lt -20.

83
Demand Revelation

• Then A suffers a GC tax of -10 50 40,
• As net payoff is - 20 - 40 -60 lt -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 Cs net valuations sum to(40 - 60) (110
  - 60) 30 gt 0.
• A, B Cs net valuations sum to
• (50 - 60) 30 20 gt 0.

86
Demand Revelation

• Assign c1 60, c2 60, c3 60.
• A Cs net valuations sum to(40 - 60) (110
  - 60) 30 gt 0.
• A, B Cs net valuations sum to
• (50 - 60) 30 20 gt 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 gt -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 gt -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 (40 - 60) (110 - 60) lt 0I.e. B must
  state sB lt -30.

90
Demand Revelation

• Then B suffers a GC tax of -20 50 30,
• Bs net payoff is - 10 - 30 -40 lt -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 Bs net valuations sum to(40 - 60) (50 -
  60) -30 lt 0.
• A, B Cs net valuations sum to
• (110 - 60) - 30 20 gt 0.

93
Demand Revelation

• Assign c1 60, c2 60, c3 60.
• A Bs net valuations sum to(40 - 60) (50 -
  60) -30 lt 0.
• A, B Cs net valuations sum to
• (110 - 60) - 30 20 gt 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 gt 50 changes nothing. C stays pivotal and
  must pay a GC tax of -(40 - 60) - (50 - 60)
  30, for a net payoff of (110 - 60) - 30 20
  gt 0.

96
Demand Revelation

• What net valuation sC should C state?
• sC gt 50 changes nothing. C stays pivotal and
  must pay a GC tax of -(40 - 60) - (50 - 60)
  30, for a net payoff of (110 - 60) - 30 20
  gt 0.
• sC lt 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 gt 50 changes nothing. C stays pivotal and
  must pay a GC tax of -(40 - 60) - (50 - 60)
  30, for a net payoff of (110 - 60) - 30 20
  gt 0.
• sC lt 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.