Public Goods 2
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Transcript of Public Goods 2
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Variable Public Good Quantities
• E.g. how many broadcast TV programs, or how much land to include into a national park.
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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.
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Variable Public Good Quantities
• Budget allocations must satisfyx x c G w wA B A B ( ) .
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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
x x c G w wA B A B ( ) .
MRS MRS MCA B ( ).G
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Variable Public Good Quantities
• Pareto efficiency condition for public good supply is
• Why?MRS MRS MCA B ( ).G
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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.
MRS MRS MCA B ( ).G
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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.
MRS MRS MCA B ( ).G
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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.
MRS MRSA B
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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.
MRS MRS MCA B ( )G
MRS MRSA B
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Variable Public Good Quantities
• Now suppose MRS MRS MCA B ( ).G
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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.
MRS MRS MCA B ( ).GMRS MRSA B
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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.
MRS MRS MCA B ( ).GMRS MRSA B
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Variable Public Good Quantities
• Hence, necessarily, efficient public good production requires
MRS MRS MCA B ( ).G
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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
MRS MRS MCA B ( ).G
MRS MCii
nG
1
( ).
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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?
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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.
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Demand Revelation
• ni = vi - ci is net value, for i = 1,…,N.• Pareto-improving to supply the public
good if
v ci ii
N
i
N
11
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Demand Revelation
• ni = vi - ci is net value, for i = 1,…,N.• Pareto-improving to supply the public
good if
v c ni i ii
N
i
N
i
N
0
111.
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Demand Revelation
• If and
or and
then individual j is pivotal; i.e. changes the supply decision.
nii j
N
0 n ni j
i j
N
0
nii j
N
0 n ni j
i j
N
0
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Demand Revelation
• What loss does a pivotal individual j inflict on others?
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Demand Revelation
• What loss does a pivotal individual j inflict on others?
• If then is the loss.nii j
N
0,
nii j
N0
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Demand Revelation
• What loss does a pivotal individual j inflict on others?
• If then is the loss.
• If then is the loss.
nii j
N
0,
nii j
N0
nii j
N
0, ni
i j
N
0
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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.
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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.si
i
N
01
;
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Demand Revelation
• A pivotal person j who changes the outcome from supply to not supply
pays a tax of sii j
N.
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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
sii j
N.
sii j
N.
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Demand Revelation
• Note: Taxes are not paid to other individuals, but to some other agent outside the market.
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Demand Revelation
• Why is the GC tax scheme a revelation mechanism?
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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.
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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.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• B & C’s net valuations sum to
$(50 - 60) + $(110 - 60) = $40 > 0.• A, B & C’s net valuations sum to• $(40 - 60) + $40 = $20 > 0.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• B & C’s net valuations sum to
$(50 - 60) + $(110 - 60) = $40 > 0.• A, B & C’s net valuations sum to• $(40 - 60) + $40 = $20 > 0.• So A is not pivotal.
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Demand Revelation• If B and C are truthful, then what net
valuation sA should A state?
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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.
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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 + $(50 - 60) + $(110 - 60) < 0;I.e. A must state sA < -$40.
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Demand Revelation• Then A suffers a GC tax of
-$10 + $50 = $40,• A’s net payoff is
- $20 - $40 = -$60 < -$20.
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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.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• A & C’s net valuations sum to
$(40 - 60) + $(110 - 60) = $30 > 0.• A, B & C’s net valuations sum to• $(50 - 60) + $30 = $20 > 0.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• A & C’s net valuations sum to
$(40 - 60) + $(110 - 60) = $30 > 0.• A, B & C’s net valuations sum to• $(50 - 60) + $30 = $20 > 0.• So B is not pivotal.
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Demand Revelation• What net valuation sB should B state?
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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.
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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 + $(40 - 60) + $(110 - 60) < 0;I.e. B must state sB < -$30.
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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.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• A & B’s net valuations sum to
$(40 - 60) + $(50 - 60) = -$30 < 0.• A, B & C’s net valuations sum to• $(110 - 60) - $30 = $20 > 0.
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Demand Revelation
• Assign c1 = $60, c2 = $60, c3 = $60.• A & B’s net valuations sum to
$(40 - 60) + $(50 - 60) = -$30 < 0.• A, B & C’s net valuations sum to• $(110 - 60) - $30 = $20 > 0.• So C is pivotal.
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Demand Revelation• What net valuation sC should C state?
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Demand Revelation• What net valuation sC should C state?
• sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
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Demand Revelation• What net valuation sC should C state?
• sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
• sC < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.
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Demand Revelation• What net valuation sC should C state?
• sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $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.
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Demand Revelation• GC tax scheme implements efficient
supply of the public good.
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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.