Chapter Thirty-Six
Public Goods
1
Important Characteristics of Goods
u
A good is excludable if a person can be
prevented from using it.
– Excludable: fish tacos, wireless Internet
access
– Not excludable: FM radio signals, national
defense
u
A good is rival in consumption if one person’s
use of it diminishes others’ use.
– Rival: fish tacos
– Not rival:
An MP3 file of Kanye West’s latest single
2
The Different Kinds of Goods
Private goods: excludable, rival in consumption
Example: food
Public goods: not excludable, not rival
Example: national defense
Street light
Public park
Common resources: rival but not excludable
Example: fish in the ocean
Club goods: excludable but not rival
Example: cable TV
3
ACTIVE LEARNING
Categorizing roads
u
u
1
A road is which of the four kinds of
goods?
Hint: The answer depends on whether
the road is congested or not, and
whether it’s a toll road or not. Consider
the different cases.
4
ACTIVE LEARNING
Answers
u
u
1
Rival in consumption? Only if congested.
Excludable? Only if a toll road.
Four possibilities:
Uncongested non-toll road: public good
Uncongested toll road: club good
Congested non-toll road: common resource
Congested toll road: private good
5
Public Goods -- Examples
u
u
u
u
u
Broadcast radio and TV programs.
National defense.
Public highways.
Reductions in air pollution.
National parks.
6
Reservation Prices
u
u
u
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 ).
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Reservation Prices
u
u
u
u
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).
8
Reservation Prices
u
u
u
u
u
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).
9
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
10
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
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When Should a Public Good Be
Provided?
u
u
u
u
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.
12
When Should a Public Good Be
Provided?
u
Payments must be individually
rational; i.e.
U A ( wA ,0 ) ≤ U A ( wA − gA ,1)
and
UB ( wB ,0 ) ≤ UB ( wB − gB ,1).
13
When Should a Public Good Be
Provided?
u
Payments must be individually
rational; i.e.
U A ( wA ,0 ) ≤ U A ( wA − gA ,1)
and
UB ( wB ,0 ) ≤ UB ( wB − gB ,1).
u
Therefore, necessarily
gA ≤ rA and gB ≤ r B .
14
When Should a Public Good Be
Provided?
u
And if U A ( wA ,0 ) < U A ( wA − gA ,1)
and
UB ( wB ,0 ) < UB ( wB − gB ,1)
then it is Pareto-improving to supply
the unit of good
15
When Should a Public Good Be
Provided?
u
And if U A ( wA ,0 ) < U A ( wA − gA ,1)
and
UB ( wB ,0 ) < UB ( wB − gB ,1)
then it is Pareto-improving to supply
the unit of good, so rA + r B > c
is sufficient for it to be efficient to
supply the good.
16
Private Provision of a Public
Good?
u
u
u
Suppose rA > c and r B < c .
Then A would supply the good even
if B made no contribution.
B then enjoys the good for free; freeriding.
17
Private Provision of a Public
Good?
u
u
Suppose rA < c and r B < c .
Then neither A nor B will supply the
good alone.
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Private Provision of a Public
Good?
u
u
u
Suppose rA < c and r B < c .
Then neither A nor B will supply the
good alone.
Yet, if rA + r B > c also, then it is Paretoimproving for the good to be supplied.
19
Private Provision of a Public
Good?
u
u
u
u
Suppose rA < c and r B < c .
Then neither A nor B will supply the
good alone.
Yet, if rA + r B > 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.
20
Free-Riding
u
u
u
u
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.
21
Free-Riding
u
u
u
u
u
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.
22
Free-Riding
Player B
Don’t
Buy
Buy
Buy -$20, -$35 -$20, $65
Player A
Don’t
$80, -$35
Buy
$0, $0
If A swich to
buy, payoff <0;
if B switch to
buy, payoff<0,
so they both
don’t have
incentive to
change
23
Free-Riding
Player B
Don’t
Buy
Buy
Buy -$20, -$35 -$20, $65
Player A
Don’t
$80, -$35
Buy
$0, $0
(Don’t’ Buy, Don’t Buy) is the unique NE.
24
Free-Riding
Player B
Don’t
Buy
Buy
Buy -$20, -$35 -$20, $65
Player A
Don’t
$80, -$35
Buy
$0, $0
But (Don’t’ Buy, Don’t Buy) is inefficient.
25
Free-Riding
u
u
u
u
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.
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Free-Riding
Player B
Don’t
Contribute Contribute
Contribute
Player A
Don’t
Contribute
$20, $25
-$60, $0
$0, -$40
$0, $0
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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
28
(Don’t Contribute, Don’t Contribute).
Free-Riding
u
u
u
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.
29
Variable Public Good Quantities
u
E.g. how many broadcast TV
programs, or how much land to
include into a national park.
30
Variable Public Good Quantities
u
u
u
u
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.
31
Variable Public Good Quantities
u
Budget allocations must satisfy
xA + xB + c (G ) = wA + wB .
32
Variable Public Good Quantities
u
u
u
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 + MRS B = MC (G ).
33
Variable Public Good Quantities
u
u
Pareto efficiency condition for public
good supply is
MRS A + MRS B = MC (G ).
Why?
34
Variable Public Good Quantities
u
u
u
Pareto efficiency condition for public
good supply is
MRS A + MRS B = 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.
35
Variable Public Good Quantities
u
u
u
Suppose MRS A + MRS B < MC (G ).
MRSA is A’s utility-preserving
compensation in private good units
for a one-unit reduction in public
good.
Similarly for B.
36
Variable Public Good Quantities
u
MRS A + MRS B is the total payment to
A & B of private good that preserves
both utilities if G is lowered by 1 unit.
37
Variable Public Good Quantities
u
u
MRS A + MRS B 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.
38
Variable Public Good Quantities
u
Now suppose MRS A + MRS B > MC (G ).
39
Variable Public Good Quantities
u
u
Now suppose MRS A + MRS B > 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.
40
Variable Public Good Quantities
u
u
u
Now suppose MRS A + MRS B > 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.
41
Variable Public Good Quantities
u
Hence, necessarily, efficient public
good production requires
MRS A + MRS B = MC (G ).
42
Variable Public Good Quantities
u
u
Hence, necessarily, efficient public
good production requires
MRS A + MRS B = MC (G ).
Suppose there are n consumers; i =
1,…,n. Then efficient public good
production requires
n
∑ MRS i = MC ( G ).
i =1
43
Efficient Public Good Supply -the Quasilinear Preferences Case
u
u
Two consumers, A and B.
U i ( xi , G ) = xi + fi (G ); i = A , B .
44
Efficient Public Good Supply -the Quasilinear Preferences Case
u
u
u
u
Two consumers, A and B.
U i ( xi , G ) = xi + fi (G ); i = A , B .
MRSi = − fi′(G ); i = A , B .
Utility-maximization requires
pG
⇒ fi′(G ) = pG ; i = A , B .
MRSi = −
px
45
Efficient Public Good Supply -the Quasilinear Preferences Case
u
u
u
u
Two consumers, A and B.
U i ( xi , G ) = xi + fi (G ); i = A , B .
MRSi = − fi′(G ); i = A , B .
=MU(G)
Utility-maximization requires
pG
⇒ fi′(G ) = pG ; i = A , B .
MRSi = −
px
u
pG = fi′(G ) is i’s public good
demand/marg. utility curve; i = A,B.
46
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUB
Equals to MRS
MUA
G
47
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MUB
MUA
G
48
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MUB
MC(G)
MUA
G
49
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MC(G)
MUB
MUA
G*
G
50
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
MUA+MUB
MC(G)
MUB
pG*
MUA
G*
G
51
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
*
pG = MU A (G*) + MUB (G*)
MUA+MUB
MC(G)
MUB
pG*
MUA
G*
G
52
Efficient Public Good Supply -the Quasilinear Preferences Case
pG
*
pG = MU A (G*) + MUB (G*)
MUA+MUB
MUB
MC(G)
pG*
MUA
G*
G
Efficient public good supply requires A & B
53
to state truthfully their marginal valuations.
Free-Riding Revisited
u
When is free-riding individually
rational?
54
Free-Riding Revisited
u
u
When is free-riding individually
rational?
Individuals can contribute only
positively to public good supply;
nobody can lower the supply level.
55
Free-Riding Revisited
u
u
u
u
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.
56
Free-Riding Revisited
u
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.
57
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gA
xB
58
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB > 0
gA
gB < 0 is not allowed
xB
59
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB > 0
gA
gB < 0 is not allowed
xB
60
Free-Riding Revisited
G
B’s budget constraint; slope = -1
gB > 0
gA
gB < 0 is not allowed
xB
61
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
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Skip 36.7 – 36.11
63