Chapter Thirty-FivePublic Goods整理课件Public Goods - DefinitionuA 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.整理课件Public Goods - ExamplesuBroadcast radio and TV programs.uNational defense.uPublic highways.uReductions in air pollution.uNational parks.整理课件Reservation PricesuA consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.uConsumers wealth is uUtility of not having the good is整理课件Reservation PricesuA consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.uConsumers wealth is uUtility of not having the good isuUtility of paying p for the good is整理课件Reservation PricesuA consumers reservation price for a unit of a good is his maximum willingness-to-pay for it.uConsumers wealth is uUtility of not having the good isuUtility of paying p for the good isuReservation price r is defined by整理课件Reservation Prices; An ExampleConsumers utility isUtility of not buying a unit of good 2 isUtility of buying one unit of good 2 atprice p is整理课件Reservation Prices; An ExampleReservation price r is defined byI.e. by整理课件When Should a Public Good Be Provided?uOne unit of the good costs c.uTwo consumers, A and B.uIndividual payments for providing the public good are gA and gB.ugA + gB c if the good is to be provided.整理课件When Should a Public Good Be Provided?uPayments must be individually rational; i.e.and整理课件When Should a Public Good Be Provided?uPayments must be individually rational; i.e.anduTherefore, necessarily and整理课件When Should a Public Good Be Provided?uAnd ifandthen it is Pareto-improving to supply the unit of good整理课件When Should a Public Good Be Provided?uAnd ifandthen it is Pareto-improving to supply the unit of good, so is sufficient for it to be efficient to supply the good.整理课件Private Provision of a Public Good?uSuppose and .uThen A would supply the good even if B made no contribution.uB then enjoys the good for free; free-riding.整理课件Private Provision of a Public Good?uSuppose and .uThen neither A nor B will supply the good alone.整理课件Private Provision of a Public Good?uSuppose and .uThen neither A nor B will supply the good alone.uYet, if also, then it is Pareto-improving for the good to be supplied.整理课件Private Provision of a Public Good?uSuppose and .uThen neither A nor B will supply the good alone.uYet, if also, then it is Pareto-improving for the good to be supplied.uA and B may try to free-ride on each other, causing no good to be supplied.整理课件Free-RidinguSuppose A and B each have just two actions - individually supply a public good, or not.uCost of supply c = $100.uPayoff to A from the good = $80.uPayoff to B from the good = $65.整理课件Free-RidinguSuppose A and B each have just two actions - individually supply a public good, or not.uCost of supply c = $100.uPayoff to A from the good = $80.uPayoff to B from the good = $65.u$80 + $65 $100, so supplying the good is Pareto-improving.整理课件Free-RidingBuyDontBuyBuyDontBuyPlayer APlayer B整理课件Free-RidingBuyDontBuyBuyDontBuyPlayer APlayer B(Dont Buy, Dont Buy) is the unique NE.整理课件Free-RidingBuyDontBuyBuyDontBuyPlayer APlayer BBut (Dont Buy, Dont Buy) is inefficient.整理课件Free-RidinguNow allow A and B to make contributions to supplying the good.uE.g. A contributes $60 and B contributes $40.uPayoff to A from the good = $40 $0.uPayoff to B from the good = $25 $0.整理课件Free-RidingContributeDontContributeContributeDontContributePlayer APlayer B整理课件Free-RidingContributeDontContributeContributeDontContributePlayer APlayer BTwo NE: (Contribute, Contribute) and (Dont Contribute, Dont Contribute).整理课件Free-RidinguSo allowing contributions makes possible supply of a public good when no individual will supply the good alone.uBut what contribution scheme is best?uAnd free-riding can persist even with contributions.整理课件Variable Public Good QuantitiesuE.g. how many broadcast TV programs, or how much land to include into a national park.整理课件Variable Public Good QuantitiesuE.g. how many broadcast TV programs, or how much land to include into a national park.uc(G) is the production cost of G units of public good.uTwo individuals, A and B.uPrivate consumptions are xA, xB.整理课件Variable Public Good QuantitiesuBudget allocations must satisfy整理课件Variable Public Good QuantitiesuBudget allocations must satisfyuMRSA & MRSB are A & Bs marg. rates of substitution between the private and public goods.uPareto efficiency condition for public good supply is整理课件Variable Public Good QuantitiesuPareto efficiency condition for public good supply isuWhy?整理课件Variable Public Good QuantitiesuPareto efficiency condition for public good supply isuWhy?uThe public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B.整理课件Variable Public Good QuantitiesuSupposeuMRSA is As utility-preserving compensation in private good units for a one-unit reduction in public good.uSimilarly for B.整理课件Variable Public Good Quantitiesu is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.整理课件Variable Public Good Quantitiesu is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.uSince , making 1 less public good unit releases more private good than the compensation payment requires Pareto-improvement from reduced G.整理课件Variable Public Good QuantitiesuNow suppose整理课件Variable Public Good QuantitiesuNow supposeu is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.整理课件Variable Public Good QuantitiesuNow supposeu is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.uThis payment provides more than 1 more public good unit Pareto-improvement from increased G.整理课件Variable Public Good QuantitiesuHence, necessarily, efficient public good production requires整理课件Variable Public Good QuantitiesuHence, necessarily, efficient public good production requiresuSuppose there are n consumers; i = 1,n. Then efficient public good production requires整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseuTwo consumers, A and B.u 整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseuTwo consumers, A and B.u u uUtility-maximization requires整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseuTwo consumers, A and B.u u uUtility-maximization requiresu is is public good demand/marg. utility curve; i = A,B. 整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseMUAMUBpGG整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseMUAMUBMUA+MUBpGG整理课件Efficient Public Good Supply - the Quasilinear Preferences CasepGMUAMUBMUA+MUBMC(G)G整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*pG*整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*pG*整理课件Efficient Public Good Supply - the Quasilinear Preferences CaseGpGMUAMUBMUA+MUBMC(G)G*pG*Efficient public good supply requires A & Bto state truthfully their marginal valuations.整理课件Free-Riding RevisiteduWhen is free-riding individually rational?整理课件Free-Riding RevisiteduWhen is free-riding individually rational?uIndividuals can contribute only positively to public good supply; nobody can lower the supply level.整理课件Free-Riding RevisiteduWhen is free-riding individually rational?uIndividuals can contribute only positively to public good supply; nobody can lower the supply level.uIndividual utility-maximization may require a lower public good level.uFree-riding is rational in such cases.整理课件Free-Riding RevisiteduGiven A contributes gA units of public good, Bs problem issubject to整理课件Free-Riding RevisitedGxBgABs budget constraint; slope = -1整理课件Free-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowed整理课件Free-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowed整理课件Free-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowed整理课件Free-Riding RevisitedGxBgABs budget constraint; slope = -1is not allowed(i.e. free-riding) is best for B整理课件Demand RevelationuA scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a revelation mechanism.uE.g. the Groves-Clarke taxation scheme.uHow does it work?整理课件Demand RevelationuN individuals; i = 1,N.uAll have quasi-linear preferences.uvi is individual is true (private) valuation of the public good.uIndividual i must provide ci private good units if the public good is supplied.整理课件Demand Revelationuni = vi - ci is net value, for i = 1,N.uPareto-improving to supply the public good if整理课件Demand Revelationuni = vi - ci is net value, for i = 1,N.uPareto-improving to supply the public good if整理课件Demand RevelationuIf andor andthen individual j is pivotal; i.e. changes the supply decision.整理课件Demand RevelationuWhat loss does a pivotal individual j inflict on others?整理课件Demand RevelationuWhat loss does a pivotal individual j inflict on others?uIf then is the loss.整理课件Demand RevelationuWhat loss does a pivotal individual j inflict on others?uIf then is the loss.uIf then is the loss.整理课件Demand RevelationuFor efficiency, a pivotal agent must face the full cost or benefit of her action.uThe GC tax scheme makes pivotal agents face the full stated costs or benefits of their actions in a way that makes these statements truthful.整理课件Demand RevelationuThe GC tax scheme:uAssign a cost ci to each individual.uEach agent states a public good net valuation, si.uPublic good is supplied if otherwise not.整理课件Demand RevelationuA pivotal person j who changes the outcome from supply to not supplypays a tax of整理课件Demand RevelationuA pivotal person j who changes the outcome from supply to not supplypays a tax ofuA pivotal person j who changes the outcome from not supply to supplypays a tax of整理课件Demand RevelationuNote: Taxes are not paid to other individuals, but to some other agent outside the market.整理课件Demand RevelationuWhy is the GC tax scheme a revelation mechanism?整理课件Demand RevelationuWhy is the GC tax scheme a revelation mechanism?uAn example: 3 persons; A, B and C.uValuations of the public good are:$40 for A, $50 for B, $110 for C.uCost of supplying the good is $180.整理课件Demand RevelationuWhy is the GC tax scheme a revelation mechanism?uAn example: 3 persons; A, B and C.uValuations of the public good are:$40 for A, $50 for B, $110 for C.uCost of supplying the good is $180.u$180 0.uA, B & Cs net valuations sum tou$(40 - 60) + $40 = $20 0.整理课件Demand RevelationuAssign c1 = $60, c2 = $60, c3 = $60.uB & Cs net valuations sum to$(50 - 60) + $(110 - 60) = $40 0.uA, B & Cs net valuations sum tou$(40 - 60) + $40 = $20 0.uSo A is not pivotal.整理课件Demand RevelationuIf B and C are truthful, then what net valuation sA should A state?整理课件Demand RevelationuIf B and C are truthful, then what net valuation sA should A state?uIf sA -$20, then A makes supply of the public good, and a loss of $20 to him, more likely.整理课件Demand RevelationuIf B and C are truthful, then what net valuation sA should A state?uIf sA -$20, then A makes supply of the public good, and a loss of $20 to him, more likely.uA prevents supply by becoming pivotal, requiring sA + $(50 - 60) + $(110 - 60) 0;I.e. A must state sA -$40.整理课件Demand RevelationuThen A suffers a GC tax of -$10 + $50 = $40,uAs net payoff is - $20 - $40 = -$60 -$20.整理课件Demand RevelationuThen A suffers a GC tax of -$10 + $50 = $40,uAs net payoff is - $20 - $40 = -$60 0.uA, B & Cs net valuations sum tou$(50 - 60) + $30 = $20 0.整理课件Demand RevelationuAssign c1 = $60, c2 = $60, c3 = $60.uA & Cs net valuations sum to$(40 - 60) + $(110 - 60) = $30 0.uA, B & Cs net valuations sum tou$(50 - 60) + $30 = $20 0.uSo B is not pivotal.整理课件Demand RevelationuWhat net valuation sB should B state?整理课件Demand RevelationuWhat net valuation sB should B state?uIf sB -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.整理课件Demand RevelationuWhat net valuation sB should B state?uIf sB -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.uB prevents supply by becoming pivotal, requiring sB + $(40 - 60) + $(110 - 60) 0;I.e. B must state sB -$30.整理课件Demand RevelationuThen B suffers a GC tax of -$20 + $50 = $30,uBs net payoff is - $10 - $30 = -$40 -$10.uB can do no better than state the truth; sB = -$10.整理课件Demand RevelationuAssign c1 = $60, c2 = $60, c3 = $60.整理课件Demand RevelationuAssign c1 = $60, c2 = $60, c3 = $60.uA & Bs net valuations sum to$(40 - 60) + $(50 - 60) = -$30 0.整理课件Demand RevelationuAssign c1 = $60, c2 = $60, c3 = $60.uA & Bs net valuations sum to$(40 - 60) + $(50 - 60) = -$30 0.uSo C is pivotal.整理课件Demand RevelationuWhat net valuation sC should C state?整理课件Demand RevelationuWhat net valuation sC should C state?usC $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 $0.整理课件Demand RevelationuWhat net valuation sC should C state?usC $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 $0.usC $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 $0.usC $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.uC can do no better than state the truth; sC = $50.整理课件Demand RevelationuGC tax scheme implements efficient supply of the public good.整理课件Demand RevelationuGC tax scheme implements efficient supply of the public good.uBut, causes an inefficiency due to taxes removing private good from pivotal individuals.整理课件