1、Chapter Thirty-FivePublic Goods公共物品StructureDefinitionWhen to provide a public goodEfficient provisionPrivate provision: free-ridingVariable quantities of public goodEfficient amountFree-riding problemHow to provide public goods?Demand revelationPublic Goods - DefinitionA good is purely public if it

2、 is both nonexcludable (非排他性)and nonrival(非爭性) in consumption.Nonexcludable - all consumers can consume the good.Nonrival - each consumer can consume all of the good.Public Goods - ExamplesBroadcast radio and TV programs.National defense.Public highways.Reductions in air pollution.National parks.Res

3、ervation PricesA 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 isReservation PricesA consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.Consumers wealth is Utility of

4、 not having the good isUtility of paying p for the good isReservation PricesA 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 isReservation price r is defined byReservation

5、 Prices; An ExampleConsumers utility isUtility of not buying a unit of good 2 isUtility of buying one unit of good 2 atprice p isReservation Prices; An ExampleReservation price r is defined byI.e. byWhen Should a Public Good Be Provided?One unit of the good costs c.Two consumers, A and B.Individual

6、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.andWhen Should a Public Good Be Provided?Payments must be individually rational; i.e.andTherefore, necessarily andWhen Should

7、 a Public Good Be Provided?And ifandthen it is Pareto-improving to supply the unit of goodWhen Should a Public Good Be Provided?And ifandthen it is Pareto-improving to supply the unit of good, so is sufficient for it to be efficient to supply the good.Private Provision of a Public Good?Suppose and .

8、Then A would supply the good even if B made no contribution.B then enjoys the good for free; free-riding (免費搭車).Private Provision of a Public Good?Suppose and .Then neither A nor B will supply the good alone.Private Provision of a Public Good?Suppose and .Then neither A nor B will supply the good al

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

10、to be supplied.Free-RidingSuppose 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-RidingSuppose A and B each have just two actions - individually supply a public good, or not.C

11、ost 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-RidingBuyDontBuyBuyDontBuyPlayer APlayer B(Dont Buy, Dont Buy) is the unique NE.Free-RidingBuyDontBuyBuyDontBuyPlayer APlayer BBut (Dont Buy, Dont Buy

12、) is inefficient.Free-RidingNow 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.Free-RidingContributeDontContributeContributeDontContributePlayer APlayer BFree-RidingContribut

13、eDontContributeContributeDontContributePlayer APlayer BTwo NE: (Contribute, Contribute) and (Dont Contribute, Dont Contribute).Free-RidingSo 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-ridin

14、g can persist even with contributions.Variable Public Good QuantitiesE.g. TV quality, 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 Q

15、uantitiesBudget allocations must satisfyVariable Public Good QuantitiesBudget allocations must satisfyMRSA & MRSB are A & Bs marg. rates of substitution between the private and public goods.Pareto efficiency condition for public good supply isGMRSMUAMUBMUA+MUBMC(G)G*Variable Public Good QuantitiesVa

16、riable Public Good QuantitiesPareto efficiency condition for public good supply isWhy?Variable Public Good QuantitiesPareto efficiency condition for public good supply isWhy?The public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B.Variable Public G

17、ood QuantitiesSupposeMRSA is As utility-preserving compensation in private good units for a one-unit reduction in public good.Similarly for B.Variable Public Good Quantities is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.Variable Public Good Qua

18、ntities 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.Variable Public Good QuantitiesNow supposeVariable

19、Public Good QuantitiesNow suppose is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.Variable Public Good QuantitiesNow 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

20、 more than 1 more public good unit Pareto-improvement from increased G.Variable Public Good QuantitiesHence, necessarily, efficient public good production requiresVariable Public Good QuantitiesHence, necessarily, efficient public good production requiresSuppose there are n consumers; i = 1,n. Then

21、efficient public good production requiresEfficient Public Good Supply - the Quasilinear Preferences CaseTwo consumers, A and B. Efficient Public Good Supply - the Quasilinear Preferences CaseTwo consumers, A and B. Utility-maximization requiresEfficient Public Good Supply - the Quasilinear Preferenc

22、es CaseTwo consumers, A and B. Utility-maximization requires is is public good demand/marg. utility curve; i = A,B. Efficient Public Good Supply - the Quasilinear Preferences CaseMUAMUBpGGEfficient Public Good Supply - the Quasilinear Preferences CaseMUAMUBMUA+MUBpGGEfficient Public Good Supply - th

23、e Quasilinear Preferences CasepGMUAMUBMUA+MUBMC(G)GEfficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*pG*G* is the unique amount of public good at every efficient allocation.Efficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*pG*Ef

24、ficient public good supply requires A & Bto state truthfully their marginal valuations.Free-Riding RevisitedWhen 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 lowe

25、r public good level.Free-riding is rational in such cases.Free-Riding RevisitedGiven A contributes gA units of public good, Bs problem issubject toFree-Riding RevisitedGxBgABs budget constraint; slope = -1Free-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowedFree-Riding RevisitedGx

26、BgABs budget constraint; slope = -1is not allowedFree-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowedFree-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowed(i.e. free-riding) is best for BHow Else to Provide Public Goods?Command mechanismVotingParadox of votingMa

27、jority Votingx, y, z denote different economic states.3 agents; Bill, Bertha and Bob.Use simple majority voting to decide a state?More preferredLess preferredMajority VotingMajority Vote Resultsx beats yy beats zz beats xMajority voting doesnot always aggregatetransitive individualpreferences into a

28、transitive socialpreference.Nosociallybestalternative!Majority VotingRank-Order Votingx-score = 6y-score = 6z-score = 6Nostate isselected!Rank-order votingis indecisive in thiscase.Rank-order vote results(low score wins).Manipulating PreferencesAs well, most voting schemes are manipulable.I.e. one i

29、ndividual can cast an “untruthful” vote to improve the social outcome for himself.Again consider rank-order voting.Manipulating PreferencesThese are truthfulpreferences.Manipulating PreferencesThese are truthful preferences.Bob introduces anew alternativeManipulating PreferencesThese are truthfulpre

30、ferences.Bob introduces anew alternative andthen lies.Manipulating PreferencesThese are truthfulpreferences.Bob introduces anew alternative andthen lies.Rank-order voteresults.x-score = 8y-score = 7z-score = 6-score = 9z wins!Demand RevelationA scheme that makes it rational for individuals to reveal

31、 truthfully their private valuations of a public good is a revelation mechanism.E.g. the Groves-Clarke taxation scheme.How does it work?Demand RevelationN individuals; i = 1,N.All have quasi-linear preferences.vi is individual is true (private) valuation of the public good.Individual i must provide

32、ci private good units if the public good is supplied.Demand Revelationni = vi - ci is net value, for i = 1,N.Pareto-improving to supply the public good ifDemand Revelationni = vi - ci is net value, for i = 1,N.Pareto-improving to supply the public good ifDemand RevelationIf andor andthen individual

33、j is pivotal; i.e. changes the supply decision.Demand RevelationWhat loss does a pivotal individual j inflict on others?Demand RevelationWhat loss does a pivotal individual j inflict on others?If then is the loss.Demand RevelationWhat loss does a pivotal individual j inflict on others?If then is the

34、 loss.If then is the loss.Demand RevelationFor 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 RevelationThe GC tax scheme:

35、Assign a cost ci to each individual.Each agent states a public good net valuation, si.Public good is supplied if otherwise not.Demand RevelationA pivotal person j who changes the outcome from supply to not supplypays a tax ofDemand RevelationA pivotal person j who changes the outcome from supply to

36、not supplypays a tax ofA pivotal person j who changes the outcome from not supply to supplypays a tax ofDemand RevelationNote: Taxes are not paid to other individuals, but to some other agent outside the market.Demand RevelationWhy is the GC tax scheme a revelation mechanism?An example: 3 persons; A

37、, 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.$180 0.A, B & Cs net valuations sum to$(40 - 60) + $40 = $20 0.So A is not pivotal.Demand RevelationIf 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.A prevents supply by becoming pivotal, requiring sA + $(50 - 60) + $(110 - 60) 0