Chapter 36 Public Goods
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Transcript of Chapter 36 Public Goods
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Chapter 36Public Goods
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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.
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Public Goods -- Examples
Broadcast radio and TV programs. National defense. Public highways. Reductions in air pollution. National parks.
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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
U w( , ).0w.
U w p( , ). 1
U w U w r( , ) ( , ).0 1
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Reservation Prices: An ExampleConsumer’s utility isU x x x x( , ) ( ).1 2 1 2 1
Utility of not buying a unit of good 2 is
V wwp
wp
( , ) ( ) .0 0 11 1
Utility of buying one unit of good 2 atprice p is
V w pw pp
w pp
( , ) ( )( )
.
1 1 12
1 1
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Reservation Prices; An ExampleReservation price r is defined by
V w V w r( , ) ( , )0 1 I.e. by
wp
w rp
rw
1 1
22
( )
.
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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.
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When Should a Public Good Be Provided? Payments must be individually
rational; i.e.
and
U w U w gA A A A A( , ) ( , )0 1
U w U w gB B B B B( , ) ( , ).0 1
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When Should a Public Good Be Provided? Payments must be individually
rational; i.e.
and
Therefore, necessarily and
U w U w gA A A A A( , ) ( , )0 1
U w U w gB B B B B( , ) ( , ).0 1
g rA A g rB B .
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When Should a Public Good Be Provided? And if
and
then it is Pareto-improving to supply the unit of good
U w U w gA A A A A( , ) ( , )0 1
U w U w gB B B B B( , ) ( , )0 1
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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.
U w U w gA A A A A( , ) ( , )0 1
U w U w gB B B B B( , ) ( , )0 1
r r cA B
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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.
r cA r cB
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Private Provision of a Public Good?
Suppose and . Then neither A nor B will supply the
good alone.
r cA r cB
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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.
r cA r cB
r r cA B
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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.
r cA r cB
r r cA B
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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.
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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.
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Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
Don’tBuy
BuyDon’tBuy
Player A
Player B
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Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
Don’tBuy
BuyDon’tBuy
Player A
Player B
(Don’t’ Buy, Don’t Buy) is the unique NE.19
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Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
Don’tBuy
BuyDon’tBuy
Player A
Player B
But (Don’t’ Buy, Don’t Buy) is inefficient.20
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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.
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Free-Riding
$20, $25 -$60, $0
$0, -$40 $0, $0
Contribute
Don’tContribute
ContributeDon’tContribute
Player A
Player B
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Free-Riding
$20, $25 -$60, $0
$0, -$40 $0, $0
Contribute
Don’tContribute
ContributeDon’tContribute
Player A
Player B
Two NE: (Contribute, Contribute) and (Don’t Contribute, Don’t Contribute).23
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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.
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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 ( ).G
MRS 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 ( ).G
MRS 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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Efficient Public Good Supply -- the Quasilinear Preferences Case Two consumers, A and B. U x G x f G ii i i i( , ) ( ); , . A B
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Efficient Public Good Supply -- the Quasilinear Preferences Case Two consumers, A and B. Utility-maximization requires
U x G x f G ii i i i( , ) ( ); , . A BMRS f G ii i ( ); , .A B
MRSpp
f G p iiG
xi G ( ) ; , .A B
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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.
U x G x f G ii i i i( , ) ( ); , . A BMRS f G ii i ( ); , .A B
MRSpp
f G p iiG
xi G ( ) ; , .A B
p f GG i ( )
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Efficient Public Good Supply -- the Quasilinear Preferences Case
MUA
MUB
pG
G
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Efficient Public Good Supply -- the Quasilinear Preferences Case
MUA
MUB
MUA+MUB
pG
G
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Efficient Public Good Supply -- the Quasilinear Preferences CasepG
MUA
MUB
MUA+MUB
MC(G)
G
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Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
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Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
pG*
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Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
pG*
p MU G MU GG* ( *) ( *) A B
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Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
pG*
p MU G MU GG* ( *) ( *) A B
Efficient public good supply requires A & Bto state truthfully their marginal valuations.48
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Free-Riding Revisited When is free-riding individually
rational?
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Free-Riding Revisited When is free-riding individually
rational? Individuals can contribute only
positively to public good supply; nobody can lower the supply level.
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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.
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Free-Riding Revisited
Given A contributes gA units of public good, B’s problem is
subject to
max,x gB B
U x g gB B A B( , )
x g w gB B B B , .0
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Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
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Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0is not allowed
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Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0is not allowed
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Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0is not allowed
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Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0is not allowedgB 0(i.e. free-riding) is best for B
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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 ifv ci i
i
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 ifv c ni i i
i
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.ni
i 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.
sii
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.
97