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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 consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isU w( , ).0
w.
Reservation Prices
A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isUtility of paying p for the good is
U w( , ).0w.
U w p( , ).1
Reservation Prices
A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of not having the good isUtility 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 ExampleConsumers utility is U x x x x( , ) ( ).1 2 1 2 1Utility 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 1I.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 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, necessarilyand
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
DontBuy
BuyDontBuy
Player A
Player B
Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
DontBuy
BuyDontBuy
Player A
Player B
(Dont Buy, Dont Buy) is the unique NE.
Free-Riding
-$20, -$35 -$20, $65
$100, -$35 $0, $0
Buy
DontBuy
BuyDontBuy
Player A
Player B
But (Dont Buy, Dont 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
DontContribute
ContributeDontContribute
Player A
Player B
Free-Riding
$20, $25 -$60, $0
$0, -$40 $0, $0
Contribute
DontContribute
ContributeDontContribute
Player A
Player B
Two NE: (Contribute, Contribute) and(Dont Contribute, Dont 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 & Bs 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
SupposeMRSA is As 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 supposeis 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 supposeis 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 ( ); , .A B
MRS pp
f G p ii Gx
i G( ) ; , .A B
Efficient Public Good Supply -- the Quasilinear Preferences Case
Two consumers, A and B.
Utility-maximization requires
is is 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
MRS pp
f G p ii Gx
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
