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Chapter Thirty-Five. Public Goods. 35: Public Goods 36: Asymmetric Information 17: Auctions 33: Law & Economics 34: Information Technology 31: Welfare. Public Goods -- Definition. A good is purely public if it is both nonexcludable and nonrival in consumption. - PowerPoint PPT Presentation
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Chapter Thirty-Five
Public Goods
35: Public Goods 36: Asymmetric Information 17: Auctions 33: Law & Economics 34: Information Technology 31: Welfare
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)
Public Goods -- Examples
Broadcast radio and TV programs. National defense. Public highways. Reductions in air pollution. National parks.
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.
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
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
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
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
( )
.
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.
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
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 .
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
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
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
Private Provision of a Public Good?
Suppose and . Then neither A nor B will supply the
good alone.
r cA r cB
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
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
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.
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.
Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
Don’tBuy
BuyDon’tBuy
Player A
Player B
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.
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.
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.
Free-Riding
$20, $25 -$60, $0
$0, -$40 $0, $0
Contribute
Don’tContribute
ContributeDon’tContribute
Player A
Player B
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).
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.
Variable Public Good Quantities
E.g. how many broadcast TV programs, or how much land to include into a national park.
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.
Variable Public Good Quantities
Budget allocations must satisfyx x c G w wA B A B ( ) .
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
Variable Public Good Quantities
Pareto efficiency condition for public good supply is
Why?MRS MRS MCA B ( ).G
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
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
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
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
Variable Public Good Quantities
Now suppose MRS MRS MCA B ( ).G
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
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
Variable Public Good Quantities
Hence, necessarily, efficient public good production requires
MRS MRS MCA B ( ).G
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
( ).
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
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
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
Efficient Public Good Supply -- the Quasilinear Preferences Case
MUA
MUB
pG
G
Efficient Public Good Supply -- the Quasilinear Preferences Case
MUA
MUB
MUA+MUB
pG
G
Efficient Public Good Supply -- the Quasilinear Preferences Case
pG
MUA
MUB
MUA+MUB
MC(G)
G
Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
Efficient Public Good Supply -- the Quasilinear Preferences Case
G
pG
MUA
MUB
MUA+MUB
MC(G)
G*
pG*
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 -- 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.
Free-Riding Revisited
When is free-riding individually rational?
Free-Riding Revisited
When is free-riding individually rational?
Individuals can contribute only positively to public good supply; nobody can lower the supply level.
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.
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
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB 0
gB 0 is not allowed
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
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?
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.
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
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.
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
Demand Revelation
What loss does a pivotal individual j inflict on others?
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
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
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.
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
;
Demand Revelation
A pivotal person j who changes the outcome from supply to not supply
pays a tax of sii j
N.
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.
Demand Revelation
Note: Taxes are not paid to other individuals, but to some other agent outside the market.
Demand Revelation
Why is the GC tax scheme a revelation mechanism?
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.
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.
Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.
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.
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.
Demand Revelation If B and C are truthful, then what net
valuation sA should A state?
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.
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.
Demand Revelation Then A suffers a GC tax of
-$10 + $50 = $40, A’s net payoff is
- $40 < -$20.
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.
Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.
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.
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.
Demand Revelation What net valuation sB should B state?
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.
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.
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.
Demand Revelation
Assign c1 = $60, c2 = $60, c3 = $60.
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.
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.
Demand Revelation What net valuation sC should C state?
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.
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.
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.
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)
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* .
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
Demand Revelation GC tax scheme implements efficient
supply of the public good.
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.
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