ch35 public goods

8/13/2019 Ch35 Public Goods

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Chapter Thirty-Five

Public Goods


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Public Goods -- Definition

A good is purely public if it is bothnonexcludable and nonrival in

consumption. Nonexcludable -- all consumerscan consume the good.

Nonrival -- each consumer canconsume all of the good.


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Public Goods -- Examples

Broadcast radio and TV programs.National defense.

Public highways.Reductions in air pollution.National parks.


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Reservation Prices

A consumers reservation price for aunit of a good is his maximumwillingness-to-pay for it.Consumers wealth isUtility of not having the good is U w( , ).0

w.


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Reservation Prices

A consumers reservation price for aunit of a good is his maximumwillingness-to-pay for it.Consumers wealth isUtility of not having the good isUtility of paying p for the good is

U w( , ).0w.

U w p( , ).1


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Reservation Prices; An Example

Consumers utility is U x x x x ( , ) ( ).1 2 1 2 1 Utility of not buying a unit of good 2 is

V w w

p

w

p( , ) ( ) .0 0 1

1 1

Utility of buying one unit of good 2 atprice p is

V w p w p p w p p( , ) ( ) ( ) .

1 1 1 21 1


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Reservation Prices; An Example

Reservation price r is defined byV w V w r ( , ) ( , )0 1

I.e. byw p

w r p r

w

1 1

22

( ).


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When Should a Public Good BeProvided?

One unit of the good costs c.Two consumers, A and B.

Individual payments for providing thepublic good are g A and g B.g A + g B c if the good is to be

provided.


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Payments must be individuallyrational; i.e.

and U w U w g A A A A A( , ) ( , )0 1 U w U w g B B B B B( , ) ( , ).0 1


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Payments must be individuallyrational; i.e.

and

Therefore, necessarilyand

U w U w g A A A A A( , ) ( , )0 1 U w U w g B B B B B( , ) ( , ).0 1

g r A A g r B B .


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And ifand

then it is Pareto-improving to supplythe unit of good

U w U w g A A A A A( , ) ( , )0 1

U w U w g B B B B B( , ) ( , )0 1


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And ifand

then it is Pareto-improving to supplythe unit of good, sois sufficient for it to be efficient tosupply the good.

U w U w g A A A A A( , ) ( , )0 1

U w U w g B B B B B( , ) ( , )0 1

r r cA B


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Private Provision of a PublicGood?

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


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Suppose and .Then neither A nor B will supply the

good alone.

r cA r cB


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good alone. Yet, if also, then it is Pareto-improving for the good to be supplied.

r cA r cB

r r cA B


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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 eachother, causing no good to be supplied.

r cA r cB

r r cA B


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Free-Riding

Suppose A and B each have just twoactions -- individually supply a publicgood, or not.Cost of supply c = $100.Payoff to A from the good = $80.

Payoff to B from the good = $65.


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Free-Riding

Suppose A and B each have just twoactions -- individually supply a publicgood, 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 thegood is Pareto-improving.


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Free-Riding

-$20 , -$35 -$20 , $65

$100 , -$35 $0 , $0

Buy

Dont Buy

BuyDont Buy

Player A

Player B


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Free-Riding

-$20 , -$35 -$20 , $65

$100 , -$35 $0 , $0

Buy

Dont Buy

BuyDont Buy

Player A

Player B

(Dont Buy, Dont Buy) is the unique NE.


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Free-Riding

-$20 , -$35 -$20 , $65

$100 , -$35 $0 , $0

Buy

Dont Buy

BuyDont Buy

Player A

Player B

But (Dont Buy, Dont Buy) is inefficient.


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Free-Riding

Now allow A and B to makecontributions to supplying the good.

E.g. A contributes $60 and Bcontributes $40.Payoff to A from the good = $40 > $0.

Payoff to B from the good = $25 > $0.


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Free-Riding

$20 , $25 -$60 , $0

$0 , -$40 $0 , $0

Contribute

Dont Contribute

ContributeDont Contribute

Player A

Player B


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Free-Riding

$20 , $25 -$60 , $0

$0 , -$40 $0 , $0

Contribute

Dont Contribute

ContributeDont Contribute

Player A

Player B

Two NE: (Contribute, Contribute) and(Dont Contribute, Dont Contribute).


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Free-Riding

So allowing contributions makespossible supply of a public goodwhen no individual will supply thegood alone.But what contribution scheme isbest?And free-riding can persist even withcontributions.


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Variable Public Good Quantities

E.g. how many broadcast TVprograms, or how much land toinclude into a national park.


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E.g. how many broadcast TVprograms, or how much land toinclude into a national park.c(G) is the production cost of G unitsof public good.

Two individuals, A and B.Private consumptions are x A, x B.


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


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Budget allocations must satisfy

MRS A & MRS B are A & Bs marg.rates of substitution between theprivate and public goods.

Pareto efficiency condition for publicgood supply is

x x c G w wA B A B ( ) .

MRS MRS MCA B ( ).G


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Why?MRS MRS MC

A B( ).G


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Why?The public good is nonrival inconsumption, so 1 extra unit ofpublic good is fully consumed byboth A and B.

MRS MRS MCA B

( ).G


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SupposeMRS A is As utility -preserving

compensation in private good unitsfor a one-unit reduction in publicgood.Similarly for B.

MRS MRS MCA B ( ).G


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is the total payment to A & B of private good that preservesboth utilities if G is lowered by 1 unit.

MRS MRSA B


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is the total payment to A & B of private good that preservesboth utilities if G is lowered by 1 unit.Since , making1 less public good unit releases moreprivate good than the compensationpayment requires Pareto-improvement from reduced G.

MRS MRS MCA B ( )G

MRS MRSA B


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Now suppose MRS MRS MCA B ( ).G


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Now suppose is the total payment by

A & B of private good that preservesboth utilities if G is raised by 1 unit.

MRS MRS MCA B ( ).G MRS MRSA B


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Now suppose is the total payment by

A & B of private good that preservesboth utilities if G is raised by 1 unit.This payment provides more than 1more public good unit Pareto-improvement from increased G.

MRS MRS MCA B ( ).G MRS MRSA B


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Hence, necessarily, efficient publicgood production requires

MRS MRS MCA B ( ).G


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Hence, necessarily, efficient publicgood production requires

Suppose there are n consumers; i =1,,n. Then efficient public goodproduction requires

MRS MRS MCA B ( ).G

MRS MCi i

nG

1( ).

Effi i P bli G d S l


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Efficient Public Good Supply --the Quasilinear Preferences Case

Two consumers, A and B. U x G x f G i i i i i ( , ) ( ); , . A B



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Two consumers, A and B.

Utility-maximization requires

U x G x f G i i i i i ( , ) ( ); , . A B

MRS f G i i i ( ); , .A B

MRS p

p f G p i i

G

x

i G ( ) ; , .A B



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Two consumers, A and B.

Utility-maximization requires

is is public gooddemand/marg. utility curve; i = A,B.

U x G x f G i i i i i ( , ) ( ); , . A B

MRS f G i i i ( ); , .A B

MRS p

p f G p i i

G

x

i G ( ) ; , .A B

p f G G i ( )


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MUA

MUB

p G

G


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MUA

MUB

MUA+MUB

p G

G


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Efficient Public Good Supply --the Quasilinear Preferences Casep G

MUA

MUB

MUA+MUB

MC(G)

G


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G

p G

MUA

MUB

MUA+MUB

MC(G)

G*

ff


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G

p G

MUA

MUB

MUA+MUB

MC(G)

G*

p G*

ff bl d l


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G

p G

MUA

MUB

MUA+MUB

MC(G)

G*

p G*

p MU G MU G G * ( *) ( *) A B

ffi i bli G d S l


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G

p G

MUA

MUB

MUA+MUB

MC(G)

G*

p G*

p MU G MU G G * ( *) ( *) A B

Efficient public good supply requires A & B

to state truthfully their marginal valuations.


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Free-Riding Revisited

When is free-riding individuallyrational?


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Individuals can contribute onlypositively to public good supply;nobody can lower the supply level.


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Individuals can contribute onlypositively to public good supply;nobody can lower the supply level.Individual utility-maximization mayrequire a lower public good level.Free-riding is rational in such cases.


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Given A contributes g A units ofpublic good, Bs problem is

subject to

max, x g B B

U x g g B B A B( , )

x g w g B B B B, .0


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G

x B

g A

Bs budget constraint; slope = -1


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G

x B

g A

Bs budget constraint; slope = -1 g B 0

g B 0 is not allowed


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G

x B

g A


g B 0 is not allowed


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G

x B

g A


g B 0 is not allowed g B 0 (i.e. free-riding) is best for B


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Demand Revelation

A scheme that makes it rational forindividuals to reveal truthfully theirprivate valuations of a public good isa revelation mechanism .E.g. the Groves-Clarke taxationscheme.How does it work?


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Demand Revelation

N individuals; i = 1,,N. All have quasi-linear preferences.

v i is individual is true (private)valuation of the public good.Individual i must provide c i private

good units if the public good issupplied.


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Demand Revelation

n i = v i - c i is net value, for i = 1,,N. Pareto-improving to supply thepublic good if

v ci i i

N

i

N

11


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Demand Revelation

n i = v i - c i is net value, for i = 1,,N. Pareto-improving to supply thepublic good if

v c ni i i i

N

i

N

i

N 0

111.


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Demand Revelation

If and

or and

then individual j is pivotal ; i.e.changes the supply decision.

n i i j

N 0 n ni j i j

N 0

n i i j

N 0 n ni j

i j

N 0


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Demand Revelation

What loss does a pivotal individual jinflict on others?


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Demand Revelation


If then is the loss.n i i j N 0, n i

i j N 0


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Demand Revelation


If then is the loss.

If then is the loss.

n i i j N 0, n i

i j N 0

n i i j

N 0, ni i j

N 0


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Demand Revelation

For efficiency, a pivotal agent mustface the full cost or benefit of heraction.The GC tax scheme makes pivotalagents face the full stated costs orbenefits of their actions in a way thatmakes these statements truthful .


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Demand Revelation

The GC tax scheme :Assign a cost c i to each individual.

Each agent states a public good net valuation, s i.Public good is supplied if

otherwise not.

si i

N 0

1;


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Demand Revelation

A pivotal person j who changes theoutcome from supply to not supply

pays a tax of si i j

N

.


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Demand Revelation

A pivotal person j who changes theoutcome 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

si i j

N

.

si i j

N .


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Demand Revelation

Note: Taxes are not paid to otherindividuals, but to some other agentoutside the market.


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Demand Revelation

Why is the GC tax scheme arevelation mechanism?


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Demand Revelation

Why is the GC tax scheme arevelation 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.


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Demand Revelation

Why is the GC tax scheme arevelation 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 isefficient to supply the good.


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.B & Cs net valuations sum to $(50 - 60) + $(110 - 60) = $40 > 0.A, B & Cs net valuations sum to $(40 - 60) + $40 = $20 > 0.


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.B & Cs net valuations sum to $(50 - 60) + $(110 - 60) = $40 > 0.A, B & Cs net valuations sum to $(40 - 60) + $40 = $20 > 0.So A is not pivotal.


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Demand Revelation

If B and C are truthful, then what net valuation s A should A state?


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Demand Revelation

If B and C are truthful, then what net valuation s A should A state?If s A > -$20, then A makes supply of

the public good, and a loss of $20 tohim, more likely.


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Demand Revelation

If B and C are truthful, then what net valuation s A should A state?If s A > -$20, then A makes supply of

the public good, and a loss of $20 tohim, more likely.A prevents supply by becomingpivotal, requirings A + $(50 - 60) + $(110 - 60) < 0;I.e. A must state s A < -$40.


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Demand Revelation

Then A suffers a GC tax of-$10 + $50 = $40,As net payoff is

- $20 - $40 = -$60 < -$20.A can do no better than state thetruth; s A = -$20.


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.A & Cs net valuations sum to $(40 - 60) + $(110 - 60) = $30 > 0.A, B & Cs net valuations sum to $(50 - 60) + $30 = $20 > 0.


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.A & Cs net valuations sum to $(40 - 60) + $(110 - 60) = $30 > 0.A, B & Cs net valuations sum to $(50 - 60) + $30 = $20 > 0.So B is not pivotal.

d l


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Demand Revelation

What net valuation s B should B state?

d l


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Demand Revelation

What net valuation s B should B state?If s B > -$10, then B makes supply ofthe public good, and a loss of $10 to

him, more likely.

D d R l i


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Demand Revelation

What net valuation s B should B state?If s B > -$10, then B makes supply ofthe public good, and a loss of $10 to

him, more likely.B prevents supply by becomingpivotal, requirings

B + $(40 - 60) + $(110 - 60) < 0;

I.e. B must state s B < -$30.

D d R l i


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Demand Revelation

Then B suffers a GC tax of-$20 + $50 = $30,Bs net payoff is

- $10 - $30 = -$40 < -$10.B can do no better than state thetruth; s B = -$10.

D d R l i


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.

D d R l i


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.A & Bs net valuations sum to $(40 - 60) + $(50 - 60) = -$30 < 0.A, B & Cs net valuations sum to $(110 - 60) - $30 = $20 > 0.

D d R l i


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Demand Revelation

Assign c 1 = $60, c 2 = $60, c 3 = $60.A & Bs net valuations sum to $(40 - 60) + $(50 - 60) = -$30 < 0.A, B & Cs net valuations sum to $(110 - 60) - $30 = $20 > 0.So C is pivotal.

D d R l i


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Demand Revelation

What net valuation s C should C state?

D d R l ti


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Demand Revelation

What net valuation s C should C state?s C > $50 changes nothing. C stayspivotal and must pay a GC tax of

-$(40 - 60) - $(50 - 60) = $30, for a netpayoff of $(110 - 60) - $30 = $20 > $0.

D d R l ti


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Demand Revelation

What net valuation s C should C state?s C > $50 changes nothing. C stayspivotal and must pay a GC tax of

-$(40 - 60) - $(50 - 60) = $30, for a netpayoff of $(110 - 60) - $30 = $20 > $0.s C < $50 makes it less likely that thepublic good will be supplied, in whichcase C loses $110 - $60 = $50.

D d R l ti


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Demand Revelation

What net valuation s C should C state?s C > $50 changes nothing. C stayspivotal and must pay a GC tax of-$(40 - 60) - $(50 - 60) = $30, for a netpayoff of $(110 - 60) - $30 = $20 > $0.s C < $50 makes it less likely that thepublic good will be supplied, in whichcase C loses $110 - $60 = $50.C can do no better than state thetruth; s C = $50.

D d R l ti


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Demand Revelation

GC tax scheme implements efficientsupply of the public good.

D d R l ti


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Demand Revelation

GC tax scheme implements efficientsupply of the public good.But, causes an inefficiency due totaxes removing private good frompivotal individuals.

ch35 public goods

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