Chapter Thirty-Five Public Goods. Public Goods -- Definition u A good is purely public if it is both...

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 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 wwp

wp

( , ) ( ) .0 0 11 1

Utility of buying one unit of good 2 atprice p is

V w pw pp

w pp

( , ) ( )( )

.

1 1 12

1 1

Reservation Prices; An ExampleReservation price r is defined by

V w V w r( , ) ( , )0 1 I.e. by

wp

w rp

rw

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.


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


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 .


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.

MRS MRS MCA B ( ).G


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 to A & B of private good that preserves both utilities if G is lowered by 1 unit.

MRS MRSA B


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.

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 ( ).G

MRS 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 ( ).G

MRS MRSA B


Hence, necessarily, efficient public good production requires

MRS MRS MCA B ( ).G


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


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

MRSpp

f G p iiG

xi 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 BMRS f G ii i ( ); , .A B

MRSpp

f G p iiG

xi G ( ) ; , .A B

p f GG i ( )


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 B

U x g gB B A B( , )

x g w gB B B B , .0


G

xB

gA

B’s budget constraint; slope = -1


G

xB

gA


gB 0

gB 0 is not allowed


G

xB

gA


gB 0

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

0 n ni j

i j

N

0

nii j

N

0 n ni j

i j

N

0

Demand Revelation

What loss does a pivotal individual j inflict on others?

Demand Revelation


If then is the loss.nii j

N

0,

nii j

N0

Demand Revelation


If then is the loss.

If then is the loss.

nii j

N

0,

nii j

N0

nii 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.

sii

N

01

;

Demand Revelation

A pivotal person j who changes the outcome from supply to not supply

pays a tax of sii 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

sii j

N.

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?



If sA > -$20, then A makes supply of the public good, and a loss of $20 to him, more likely.



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?


If sB > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.


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?


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.

Chapter Thirty-Five Public Goods. Public Goods -- Definition u A good is purely public if it is both...

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