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Microeconomics (Public Goods Ch 36 (Varian))
Microeconomics (Public Goods Ch 36 (Varian))
Lectures 26 & 27
Apr 24 & 27, 2017
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.
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 isU 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 w wp
wp
( , ) ( ) .0 0 11 1
�
Utility of buying one unit of good 2 atprice p isV w p w p
pw pp
( , ) ( ) ( ) .� �
� �1 1 1 2
1 1
Reservation Prices; An ExampleReservation price r is defined by
V w V w r( , ) ( , )0 1 �I.e. by
wp
w rp
r w
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 t 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 1d �
U w U w gB B B B B( , ) ( , ).0 1d �
When Should a Public Good Be Provided?
�Payments must be individually rational; i.e.
and
�Therefore, necessarilyand
U w U w gA A A A A( , ) ( , )0 1d �
U w U w gB B B B B( , ) ( , ).0 1d �
g rA Ad g rB Bd .
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
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 toA & 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 toA & 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� ! ( ).GMRS 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� ! ( ).GMRS 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 � c ( ); , .A B
MRS pp
f G p iiG
xi G � � c ( ) ; , .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 � c ( ); , .A B
MRS pp
f G p iiG
xi G � � c ( ) ; , .A B
p f GG i c( )
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 BU x g gB B A B( , )�
x g w gB B B B� t, .0
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1gB ! 0
gB � 0 is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1gB ! 0
gB � 0 is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1gB ! 0
gB � 0 is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1gB ! 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�
z¦ 0 n ni j
i j
N� !
z¦ 0
nii j
N!
z¦ 0 n ni j
i j
N� �
z¦ 0
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�
z¦ 0, � !
z¦ nii 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�
z¦ 0, � !
z¦ nii j
N0
nii j
N!
z¦ 0, ni
i j
N!
z¦ 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.si
i
N!
¦ 01
;
Demand Revelation
�A pivotal person j who changes the outcome from supply to not supply
pays a tax of sii j
N.
z¦
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.
z¦
�z¦ sii 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
- $20 - $40 = -$60 < -$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.
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.
Microeconomics (Public Goods Ch 36 (Varian))
Thank You