Chapter Thirty-Five
Public Goods
公共物品
Structure
? Definition
? When to provide a public good
– Efficient provision
– Private provision,free-riding
? Variable quantities of public good
– Efficient amount
– Free-riding problem
? How to provide public goods?
? Demand revelation
Public Goods -- Definition
?A good is purely public if it is both
nonexcludable (非排他性 )and
nonrival(非争性 ) 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 Example
Consumer’s utility isU x x x x(,) ( ).1 2 1 2 1? ?
Utility of not buying a unit of good 2 is
(,0 ) (0 1 ),U w w w? ? ?
Utility of buying one unit of good 2 at
price p is
(,1 ) ( ) ( 1 1 ) 2 ( ),U w p w p w p? ? ? ? ? ?
Reservation Prices; An Example
Reservation price r is defined by
(,0 ) (,1 )U w U w r??I.e,by
2 ( ),2ww w r r? ? ? ?
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?,
i i i i i i i(,0 ) (,1 ) (,1 )iU w U w r U w g? ? ? ?
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? ?
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? ?
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 (免费搭车 ).
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’t
Buy
Buy
Don’t
Buy
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’t
Buy
Buy
Don’t
Buy
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 = $20 > $0.
?Payoff to B from the good = $25 > $0.
Free-Riding
$20,$25 -$60,$0
$0,-$40 $0,$0
Contribute
Don’t
Contribute
Contribute
Don’t
Contribute
Player A
Player B
Free-Riding
$20,$25 -$60,$0
$0,-$40 $0,$0
Contribute
Don’t
Contribute
Contribute
Don’t
Contribute
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,TV quality,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 w
A 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? ? ? ?( ),
M R S M R S MCA B? ? ( ).G
G
ii
i x
i
d x M UM R S
d G M U? ? ?
G
MRS
MRSA
MRSB
MRSA+MRSB
MC(G)
G*
Variable Public Good Quantities
Variable Public Good Quantities
?Pareto efficiency condition for public
good supply is
?Why?
M R S M R S 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.
M R S M R S 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.
M R S M R S 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.
M R S M R SA 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.
M R S M R S MCA B? ? ( )G
M R S M R SA B?
Variable Public Good Quantities
?Now suppose M R S M R S 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.
M R S M R S MCA B? ? ( ).G
M R S M R SA 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.
M R S M R S MCA B? ? ( ).G
M R S M R SA B?
Variable Public Good Quantities
?Hence,necessarily,efficient public
good production requires
M R S M R S 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
M R S M R S MCA B? ? ( ).G
M R S MCi
i
n
G
?
? ?
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 B
M R S f G ii i? ? ? ?( );,.A B
M RS pp f G p ii G
x
i 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 B
M R S f G ii i? ? ? ?( );,.A B
M RS pp f G p ii G
x
i G? ? ? ? ? ?( ) ;,.A B
p f GG i? ?( )
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*
pG*
p MU G MU GG* ( *) ( *)? ?A B
G* is the unique
amount of public
good at every
efficient
allocation.
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 & B
to state truthfully their marginal valuations.
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 g
B BU 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 ? 0is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB ? 0
gB ? 0is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB ? 0
gB ? 0is not allowed
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
How Else to Provide Public
Goods?
?Command mechanism
?Voting
–Paradox of voting
Majority Voting
?x,y,z denote different economic
states.
?3 agents; Bill,Bertha and Bob.
?Use simple majority voting to decide
a state?
Bi ll Be rtha Bo b
x y z
y z x
z x y
More preferred
Less preferred
Majority Voting
Bi ll Be rtha Bo b
x y z
y z x
z x y
Majority Vote Results
x beats y
y beats z
z beats x
Majority voting does
not always aggregate
transitive individual
preferences into a
transitive social
preference.
No
socially
best
alternative!
Majority Voting
Rank-Order Voting
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) x (3 ) y (3 )
x-score = 6
y-score = 6
z-score = 6
No
state is
selected!
Rank-order voting
is indecisive in this
case.
Rank-order vote results
(low score wins).
Manipulating Preferences
?As well,most voting schemes are
manipulable.
?I.e,one individual can cast an
“untruthful” vote to improve the
social outcome for himself.
?Again consider rank-order voting.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) x (3 ) y (3 )
These are truthful
preferences.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) ? (3) y (3 )
? (4) x (4 ) ? (4)These are truthful
preferences.
Bob introduces a
new alternative
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) ? (3 ) y (3 )
? (4) x (4 ) ? (4)These are truthful
preferences.
Bob introduces a
new alternative and
then lies.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) ? (2)
z (3) ? (3) x (3 )
? (4) x (4 ) y (4 )These are truthful
preferences.
Bob introduces a
new alternative and
then lies.
Rank-order vote
results.
x-score = 8
y-score = 7
z-score = 6
?-score = 9
z wins!!
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 c
i i
i
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 n
i i i
i
N
i
N
i
N
? ? ?
?
?
?
?
?
? 0
111
.
Demand Revelation
?If and
or and
then individual j is pivotal; i.e,
changes the supply decision.
ni
i j
N
?
?
? 0 n ni j
i j
N
? ?
?
? 0
ni
i j
N
?
?
? 0 n ni j
i j
N
? ?
?
? 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.
ni
i j
N
?
?
? 0,? ?
?
? n i
i j
N
0
Demand Revelation
?What loss does a pivotal individual j
inflict on others?
?If then is the loss.
?If then is the loss.
ni
i j
N
?
?
? 0,? ?
?
? n i
i j
N
0
ni
i 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.
si
i
N
?
?
? 0
1;
Demand Revelation
?A pivotal person j who changes the
outcome from supply to not supply
pays a tax of
si
i 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
si
i 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?
?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.
?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?
?If sA > -$20,then A makes supply of
the public good,and a loss of $20 to
him.
?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 = -$20 < 0.
?A can do no better than state the
truth; sA = -$20.
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?
?If sB > -$10,then B makes supply of
the public good,and a loss of $10 to
him.
?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 = -$20 < 0.
?B can do no better than state the
truth; sB = -$10.
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?
?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.
?But,causes an inefficiency due to
taxes removing private good from
pivotal individuals.
Public Goods
公共物品
Structure
? Definition
? When to provide a public good
– Efficient provision
– Private provision,free-riding
? Variable quantities of public good
– Efficient amount
– Free-riding problem
? How to provide public goods?
? Demand revelation
Public Goods -- Definition
?A good is purely public if it is both
nonexcludable (非排他性 )and
nonrival(非争性 ) 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 Example
Consumer’s utility isU x x x x(,) ( ).1 2 1 2 1? ?
Utility of not buying a unit of good 2 is
(,0 ) (0 1 ),U w w w? ? ?
Utility of buying one unit of good 2 at
price p is
(,1 ) ( ) ( 1 1 ) 2 ( ),U w p w p w p? ? ? ? ? ?
Reservation Prices; An Example
Reservation price r is defined by
(,0 ) (,1 )U w U w r??I.e,by
2 ( ),2ww w r r? ? ? ?
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?,
i i i i i i i(,0 ) (,1 ) (,1 )iU w U w r U w g? ? ? ?
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? ?
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? ?
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 (免费搭车 ).
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’t
Buy
Buy
Don’t
Buy
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’t
Buy
Buy
Don’t
Buy
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 = $20 > $0.
?Payoff to B from the good = $25 > $0.
Free-Riding
$20,$25 -$60,$0
$0,-$40 $0,$0
Contribute
Don’t
Contribute
Contribute
Don’t
Contribute
Player A
Player B
Free-Riding
$20,$25 -$60,$0
$0,-$40 $0,$0
Contribute
Don’t
Contribute
Contribute
Don’t
Contribute
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,TV quality,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 w
A 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? ? ? ?( ),
M R S M R S MCA B? ? ( ).G
G
ii
i x
i
d x M UM R S
d G M U? ? ?
G
MRS
MRSA
MRSB
MRSA+MRSB
MC(G)
G*
Variable Public Good Quantities
Variable Public Good Quantities
?Pareto efficiency condition for public
good supply is
?Why?
M R S M R S 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.
M R S M R S 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.
M R S M R S 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.
M R S M R SA 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.
M R S M R S MCA B? ? ( )G
M R S M R SA B?
Variable Public Good Quantities
?Now suppose M R S M R S 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.
M R S M R S MCA B? ? ( ).G
M R S M R SA 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.
M R S M R S MCA B? ? ( ).G
M R S M R SA B?
Variable Public Good Quantities
?Hence,necessarily,efficient public
good production requires
M R S M R S 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
M R S M R S MCA B? ? ( ).G
M R S MCi
i
n
G
?
? ?
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 B
M R S f G ii i? ? ? ?( );,.A B
M RS pp f G p ii G
x
i 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 B
M R S f G ii i? ? ? ?( );,.A B
M RS pp f G p ii G
x
i G? ? ? ? ? ?( ) ;,.A B
p f GG i? ?( )
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*
pG*
p MU G MU GG* ( *) ( *)? ?A B
G* is the unique
amount of public
good at every
efficient
allocation.
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 & B
to state truthfully their marginal valuations.
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 g
B BU 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 ? 0is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB ? 0
gB ? 0is not allowed
Free-Riding Revisited
G
xB
gA
B’s budget constraint; slope = -1
gB ? 0
gB ? 0is not allowed
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
How Else to Provide Public
Goods?
?Command mechanism
?Voting
–Paradox of voting
Majority Voting
?x,y,z denote different economic
states.
?3 agents; Bill,Bertha and Bob.
?Use simple majority voting to decide
a state?
Bi ll Be rtha Bo b
x y z
y z x
z x y
More preferred
Less preferred
Majority Voting
Bi ll Be rtha Bo b
x y z
y z x
z x y
Majority Vote Results
x beats y
y beats z
z beats x
Majority voting does
not always aggregate
transitive individual
preferences into a
transitive social
preference.
No
socially
best
alternative!
Majority Voting
Rank-Order Voting
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) x (3 ) y (3 )
x-score = 6
y-score = 6
z-score = 6
No
state is
selected!
Rank-order voting
is indecisive in this
case.
Rank-order vote results
(low score wins).
Manipulating Preferences
?As well,most voting schemes are
manipulable.
?I.e,one individual can cast an
“untruthful” vote to improve the
social outcome for himself.
?Again consider rank-order voting.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) x (3 ) y (3 )
These are truthful
preferences.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) ? (3) y (3 )
? (4) x (4 ) ? (4)These are truthful
preferences.
Bob introduces a
new alternative
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) x (2 )
z (3) ? (3 ) y (3 )
? (4) x (4 ) ? (4)These are truthful
preferences.
Bob introduces a
new alternative and
then lies.
Manipulating Preferences
Bi ll Bertha Bob
x (1 ) y (1 ) z (1)
y (2 ) z (2) ? (2)
z (3) ? (3) x (3 )
? (4) x (4 ) y (4 )These are truthful
preferences.
Bob introduces a
new alternative and
then lies.
Rank-order vote
results.
x-score = 8
y-score = 7
z-score = 6
?-score = 9
z wins!!
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 c
i i
i
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 n
i i i
i
N
i
N
i
N
? ? ?
?
?
?
?
?
? 0
111
.
Demand Revelation
?If and
or and
then individual j is pivotal; i.e,
changes the supply decision.
ni
i j
N
?
?
? 0 n ni j
i j
N
? ?
?
? 0
ni
i j
N
?
?
? 0 n ni j
i j
N
? ?
?
? 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.
ni
i j
N
?
?
? 0,? ?
?
? n i
i j
N
0
Demand Revelation
?What loss does a pivotal individual j
inflict on others?
?If then is the loss.
?If then is the loss.
ni
i j
N
?
?
? 0,? ?
?
? n i
i j
N
0
ni
i 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.
si
i
N
?
?
? 0
1;
Demand Revelation
?A pivotal person j who changes the
outcome from supply to not supply
pays a tax of
si
i 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
si
i 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?
?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.
?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?
?If sA > -$20,then A makes supply of
the public good,and a loss of $20 to
him.
?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 = -$20 < 0.
?A can do no better than state the
truth; sA = -$20.
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?
?If sB > -$10,then B makes supply of
the public good,and a loss of $10 to
him.
?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 = -$20 < 0.
?B can do no better than state the
truth; sB = -$10.
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?
?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.
?But,causes an inefficiency due to
taxes removing private good from
pivotal individuals.
