Microeconomics (Public Goods Ch 36 (Varian)) - web.iitd.ac.inweb.iitd.ac.in/~debasis/Lectures_HUL212(2017)/Public_goods(Varian... · Public Goods --Definition A good is purely public

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


�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


�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.


�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 �


�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 .


�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� �


�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 �


�Suppose and .�Then neither A nor B will supply the

good alone.

r cA � r cB �



good alone.�Yet, if also, then it is Pareto-

improving for the good to be supplied.

r cA � r cB �

r r cA B� !



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.


�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.


�Budget allocations must satisfyx x c G w wA B A B� � �( ) .


�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



�Why?MRS MRS MCA B� ( ).G



�Why?�The public good is nonrival in

consumption, so 1 extra unit of public good is fully consumed by both A and B.



�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


� is the total payment toA & B of private good that preserves both utilities if G is lowered by 1 unit.

MRS MRSA B�


� 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�


�Now suppose MRS MRS MCA B� ! ( ).G


�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�


�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�


�Hence, necessarily, efficient public good production requires



�Hence, necessarily, efficient public good production requires

�Suppose there are n consumers; i = 1,…,n. Then efficient public good production requires


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


�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


�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( )


MUA

MUB

pG

G


MUA

MUB

MUA+MUB

pG

G


pG

MUA

MUB

MUA+MUB

MC(G)

G


G

pG

MUA

MUB

MUA+MUB

MC(G)

G*


G

pG

MUA

MUB

MUA+MUB

MC(G)

G*

pG*


G

pG

MUA

MUB

MUA+MUB

MC(G)

G*

pG*

p MU G MU GG* ( *) ( *) �A B


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?



� Individuals can contribute only positively to public good supply; nobody can lower the supply level.



� 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.


�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


G

xB

gA

B’s budget constraint; slope = -1


G

xB

gA

B’s budget constraint; slope = -1gB ! 0

gB � 0 is not allowed


G

xB

gA




G

xB

gA




G

xB

gA


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


� If then is the loss.nii j

N�

z¦ 0, � !

z¦ nii j

N0

Demand Revelation


� 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


�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


�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?


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.


valuation sA should A state?� If sA > -$20, then A makes supply of


�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


Demand Revelation�What net valuation sB should B state?� If sB > -$10, then B makes supply of


�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.



�sC < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.



�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.


Thank You

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Microeconomics (Public Goods Ch 36 (Varian)) - web.iitd.ac.inweb.iitd.ac.in/~debasis/Lectures_HUL212(2017)/Public_goods(Varian... · Public Goods --Definition A good is purely public