Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u...
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Transcript of Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u...
![Page 1: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/1.jpg)
Chapter Thirty-Five
Public Goods
![Page 2: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/2.jpg)
35: Public Goods 36: Asymmetric Information 17: Auctions 33: Law & Economics 34: Information Technology 31: Welfare
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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.
– Ausschließbarkeit (Excludability)– Konkurrenz der Güternutzung (Rivalry)
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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 U w( , ).0
w.
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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
U w( , ).0w.
U w p( , ). 1
![Page 7: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/7.jpg)
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
![Page 8: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/8.jpg)
Reservation Prices; An ExampleConsumer’s utility is U 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 p
pw pp
( , ) ( )( )
.
1 1 12
1 1
![Page 9: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/9.jpg)
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
( )
.
![Page 10: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/10.jpg)
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.
![Page 11: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/11.jpg)
When Should a Public Good Be Provided?
Payments must be individually rational; i.e.
andU 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 ifand
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 ifand
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.
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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.
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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).
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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
AA
xMRS =
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
![Page 34: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/34.jpg)
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
![Page 39: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/39.jpg)
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
![Page 40: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/40.jpg)
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
( ) ; , .A BG
i i Gx
pMRS f G p i
p
( )G ip f G
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Efficient Public Good Supply -- the Quasilinear Preferences Case
MUA
MUB
pG
G
![Page 46: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/46.jpg)
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 Case
pG
MUA
MUB
MUA+MUB
MC(G)
G
![Page 48: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/48.jpg)
Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
![Page 49: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/49.jpg)
Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
pG*
![Page 50: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/50.jpg)
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.
![Page 52: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/52.jpg)
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.
![Page 54: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/54.jpg)
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.
![Page 55: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/55.jpg)
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
![Page 56: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/56.jpg)
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
![Page 57: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/57.jpg)
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
![Page 58: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/58.jpg)
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
![Page 59: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/59.jpg)
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
![Page 60: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/60.jpg)
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowedgB 0 (i.e. free-riding) is best for B
![Page 61: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/61.jpg)
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?
![Page 62: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/62.jpg)
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.
![Page 63: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/63.jpg)
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
![Page 64: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/64.jpg)
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.
![Page 65: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/65.jpg)
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
Schlüsselperson
![Page 66: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/66.jpg)
Demand Revelation
What loss does a pivotal individual j inflict on others?
![Page 67: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/67.jpg)
Demand Revelation
What loss does a pivotal individual j inflict on others?
If then is the loss.nii j
N
0,
ni
i j
N0
![Page 68: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/68.jpg)
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,
ni
i j
N0
nii j
N
0, ni
i j
N
0
![Page 69: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/69.jpg)
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.
![Page 70: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/70.jpg)
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
;
![Page 71: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/71.jpg)
Demand Revelation
A pivotal person j who changes the outcome from supply to not supply
pays a tax of sii j
N.
![Page 72: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/72.jpg)
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.
si
i j
N.
![Page 73: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/73.jpg)
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?
![Page 75: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/75.jpg)
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.
![Page 76: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/76.jpg)
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.
![Page 77: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/77.jpg)
Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.
![Page 78: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/78.jpg)
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.
![Page 79: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/79.jpg)
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.
![Page 80: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/80.jpg)
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.
![Page 82: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/82.jpg)
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.
![Page 83: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/83.jpg)
Demand Revelation Then A suffers a GC tax of
-$10 + $50 = $40, A’s net payoff is
- $40 < -$20.
![Page 84: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/84.jpg)
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.
![Page 85: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/85.jpg)
Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.
![Page 86: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/86.jpg)
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.
![Page 87: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/87.jpg)
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.
![Page 88: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/88.jpg)
Demand Revelation What net valuation sB should B state?
![Page 89: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/89.jpg)
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.
![Page 90: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/90.jpg)
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.
![Page 91: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/91.jpg)
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.
![Page 92: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/92.jpg)
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.
![Page 94: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/94.jpg)
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.
![Page 95: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/95.jpg)
Demand Revelation What net valuation sC should C state?
![Page 96: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/96.jpg)
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.
![Page 97: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/97.jpg)
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.
![Page 98: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/98.jpg)
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.
![Page 99: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/99.jpg)
Grove-Clark Mechanism
Telling the truth is a dominant strategy(it is better to tell the truth whatever the others say)
Telling the truth is a dominant strategy(it is better to tell the truth whatever the others say)
![Page 100: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/100.jpg)
The other two say b,cThe first player true valuation is a*, but he says a.
Assume b+c > 0
1. If a*+b+c > 0 [a*> -(b+c) ]then if he calls a such that a+b+c > 0, he gets a*.
If he calls a such that a+b+c < 0then if he gets -(b+c).
But a*> -(b+c) He cannot get anything
better than telling the truth a* .
![Page 101: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/101.jpg)
The other two say b,cThe first player true valuation is a*, but he says a.
2. If a*+b+c < 0 [a*< -(b+c) ]then if he calls a such that a+b+c < 0, he gets -(b+c).
If he calls a such that a+b+c > 0then if he gets a*.
But a*< -(b+c) He cannot get anything
better than telling the truth a* .
Assume b+c < 0and a*+b+c < 0 or a*+b+c > 0
Assume b+c > 0
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Demand Revelation GC tax scheme implements efficient
supply of the public good.
![Page 103: Chapter Thirty-Five Public Goods. u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a3cb0550346a4248b5949/html5/thumbnails/103.jpg)
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.