Chapter Thirty-Five
Public Goods
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 w.
 Utility of not having the good is U ( w ,0 ).
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 w.
 Utility of not having the good is U ( w ,0 ).
 Utility of paying p for the good is
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 w.
 Utility of not having the good is U ( w ,0 ).
 Utility of paying p for the good is
U ( w  p,1).
 Reservation price r is defined by
U ( w,0 )  U ( w  r,1).
Reservation Prices; An Example
Consumer’s utility is U ( x1 , x2 )  x1 ( x2  1).
Utility of not buying a unit of good 2 is
w
w
V ( w ,0 )  ( 0  1)  .
p1
p1
Utility of buying one unit of good 2 at
price p is
w p
2( w  p )
V ( w  p,1) 
(1  1) 
.
p1
p1
Reservation Prices; An Example
Reservation price r is defined by
V ( w,0 )  V ( w  r,1)
I.e. by
w 2( w  r )
w

r .
p1
p1
2
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.
U A ( wA ,0 )  U A ( wA  gA ,1)
and
UB ( wB ,0 )  UB ( wB  gB ,1).
When Should a Public Good Be
Provided?
 Payments
must be individually
rational; i.e.
U A ( wA ,0 )  U A ( wA  gA ,1)
and
UB ( wB ,0 )  UB ( wB  gB ,1).
 Therefore,
necessarily
gA  rA and gB  rB .
When Should a Public Good Be
Provided?
if U A ( wA ,0 )  U A ( wA  gA ,1)
and
UB ( wB ,0 )  UB ( wB  gB ,1)
 And
then it is Pareto-improving to supply
the unit of good
When Should a Public Good Be
Provided?
if U A ( wA ,0 )  U A ( wA  gA ,1)
and
UB ( wB ,0 )  UB ( wB  gB ,1)
 And
then it is Pareto-improving to supply
the unit of good, so rA  rB  c
is sufficient for it to be efficient to
supply the good.
Private Provision of a Public
Good?
 c and rB  c .
 Then A would supply the good even
if B made no contribution.
 B then enjoys the good for free; freeriding.
 Suppose rA
Private Provision of a Public
Good?
 c and rB  c .
 Then neither A nor B will supply the
good alone.
 Suppose rA
Private Provision of a Public
Good?
 c and rB  c .
 Then neither A nor B will supply the
good alone.
 Yet, if rA  rB  c also, then it is Paretoimproving for the good to be supplied.
 Suppose rA
Private Provision of a Public
Good?
 c and rB  c .
 Then neither A nor B will supply the
good alone.
 Yet, if rA  rB  c also, then it is Paretoimproving for the good to be supplied.
 A and B may try to free-ride on each
other, causing no good to be supplied.
 Suppose rA
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
Player B
Don’t
Buy
Buy
Buy -$20, -$35
Player A
Don’t
Buy $100, -$35
-$20, $65
$0, $0
Free-Riding
Player B
Don’t
Buy
Buy
Buy -$20, -$35
Player A
Don’t
Buy $100, -$35
-$20, $65
$0, $0
(Don’t’ Buy, Don’t Buy) is the unique NE.
Free-Riding
Player B
Don’t
Buy
Buy
Buy -$20, -$35
Player A
Don’t
Buy $100, -$35
-$20, $65
$0, $0
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
Player B
Don’t
Contribute Contribute
Contribute
Player A
Don’t
Contribute
$20, $25
-$60, $0
$0, -$40
$0, $0
Free-Riding
Player B
Don’t
Contribute Contribute
Contribute
Player A
Don’t
Contribute
$20, $25
-$60, $0
$0, -$40
$0, $0
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 satisfy
xA  xB  c (G )  wA  wB .
Variable Public Good Quantities
 Budget
allocations must satisfy
xA  xB  c (G )  wA  wB .
 MRSA & MRSB are A & B’s marg. rates
of substitution between the private
and public goods.
 Pareto efficiency condition for public
good supply is
MRS A  MRSB  MC(G ).
Variable Public Good Quantities
 Pareto
efficiency condition for public
good supply is
MRS A  MRSB  MC(G ).
 Why?
Variable Public Good Quantities
 Pareto
efficiency condition for public
good supply is
MRS A  MRSB  MC(G ).
 Why?
 The public good is nonrival in
consumption, so 1 extra unit of
public good is fully consumed by
both A and B.
Variable Public Good Quantities
MRS A  MRSB  MC(G ).
 MRSA is A’s utility-preserving
compensation in private good units
for a one-unit reduction in public
good.
 Similarly for B.
 Suppose
Variable Public Good Quantities

MRS A  MRSB is the total payment to
A & B of private good that preserves
both utilities if G is lowered by 1 unit.
Variable Public Good Quantities
MRS A  MRSB is the total payment to
A & B of private good that preserves
both utilities if G is lowered by 1 unit.
 Since MRS A  MRS B  MC( G ), making
1 less public good unit releases more
private good than the compensation
payment requires  Paretoimprovement from reduced G.

Variable Public Good Quantities
 Now
suppose MRS A  MRSB  MC(G ).
Variable Public Good Quantities
suppose MRS A  MRSB  MC(G ).
 MRS A  MRS B is the total payment by
A & B of private good that preserves
both utilities if G is raised by 1 unit.
 Now
Variable Public Good Quantities
suppose MRS A  MRSB  MC(G ).
 MRS A  MRS B 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  Paretoimprovement from increased G.
 Now
Variable Public Good Quantities
 Hence,
necessarily, efficient public
good production requires
MRS A  MRSB  MC(G ).
Variable Public Good Quantities
 Hence,
necessarily, efficient public
good production requires
MRS A  MRSB  MC(G ).
 Suppose there are n consumers; i =
1,…,n. Then efficient public good
production requires
n
 MRS i  MC(G ).
i 1
Efficient Public Good Supply -the Quasilinear Preferences Case
 Two
consumers, A and B.
 U i ( xi , G )
 xi  fi (G ); i  A , B.
Efficient Public Good Supply -the Quasilinear Preferences Case
 Two
consumers, A and B.
 xi  fi (G ); i  A , B.
 MRS   f ( G ); i  A , B .
i
i
 Utility-maximization requires
pG
MRSi  
 fi(G )  pG ; i  A , B .
px
 U i ( xi , G )
Efficient Public Good Supply -the Quasilinear Preferences Case
 Two
consumers, A and B.
 xi  fi (G ); i  A , B.
 MRS   f ( G ); i  A , B .
i
i
 Utility-maximization requires
pG
MRSi  
 fi(G )  pG ; i  A , B .
px
 U i ( xi , G )
 fi(G ) is i’s public good
demand/marg. utility curve; i = A,B.
 pG
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUB
MUA
G
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MUB
MUA
G
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MUB
MC(G)
MUA
G
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MC(G)
MUB
MUA
G*
G
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MC(G)
MUB
pG*
MUA
G*
G
Efficient Public Good Supply -the Quasilinear Preferences Case
*
pG
 MU A (G*)  MUB (G*)
pG
MUA+MUB
MC(G)
MUB
pG*
MUA
G*
G
Efficient Public Good Supply -the Quasilinear Preferences Case
*
pG
 MU A (G*)  MUB (G*)
pG
MUA+MUB
MUB
MC(G)
pG*
MUA
G*
G
Efficient public good supply requires A & B
to 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
max UB ( xB , gA  gB )
xB , gB
subject to x B  gB  wB , gB  0.
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gA
xB
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB  0
gA
gB  0 is not allowed
xB
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB  0
gA
gB  0 is not allowed
xB
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB  0
gA
gB  0 is not allowed
xB
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gA
gB  0
gB  0 (i.e. free-riding) is best for B
gB  0 is not allowed
xB
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
= vi - ci is net value, for i = 1,…,N.
 Pareto-improving to supply the
public good if
 ni
N
N
 vi   ci
i 1
i 1
Demand Revelation
= vi - ci is net value, for i = 1,…,N.
 Pareto-improving to supply the
public good if
 ni
N
N
N
 vi   ci   ni  0.
i 1
i 1
i 1
Demand Revelation
N
N
 If  ni  0 and  ni  n j  0
i j
i j
N
N
or  ni  0 and  ni  n j  0
i j
i j
then individual j is pivotal; i.e.
changes the supply decision.
Demand Revelation
 What
loss does a pivotal individual j
inflict on others?
Demand Revelation
 What
loss does a pivotal individual j
inflict on others?
N
N
 If  ni  0, then   ni  0 is the loss.
i j
i j
Demand Revelation
 What
loss does a pivotal individual j
inflict on others?
N
N
 If  ni  0, then   ni  0 is the loss.
i j
i j
N
N
 If  ni  0, then  ni  0 is the loss.
i j
i j
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.
N
 Public good is supplied if  si  0;
i 1
otherwise not.
Demand Revelation
A
pivotal person j who changes the
outcome from supply to not supply
N
pays a tax of  si .
i j
Demand Revelation
A
pivotal person j who changes the
outcome from supply to not supply
N
pays a tax of  si .
i j
A
pivotal person j who changes the
outcome from not supply to supply
N
pays a tax of   si .
i j
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.