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Chapter Thirty-Five Public Goods

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u 35: Public Goods u 36: Asymmetric Information u 17: Auctions u 33: Law & Economics u 34: Information Technology u 31: Welfare

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Public Goods -- Definition u 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. – Ausschließbarkeit (Excludability) – Konkurrenz der Güternutzung (Rivalry)

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Public Goods -- Examples u Broadcast radio and TV programs. u National defense. u Public highways. u Reductions in air pollution. u National parks.

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Reservation Prices u A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumers wealth is u Utility of not having the good is

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Reservation Prices u A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumers wealth is u Utility of not having the good is u Utility of paying p for the good is

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Reservation Prices u A consumers reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumers wealth is u Utility of not having the good is u Utility of paying p for the good is u Reservation price r is defined by

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Reservation Prices; An Example Consumers utility is Utility of not buying a unit of good 2 is Utility of buying one unit of good 2 at price p is

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Reservation Prices; An Example Reservation price r is defined by I.e. by

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When Should a Public Good Be Provided? u One unit of the good costs c. u Two consumers, A and B. u Individual payments for providing the public good are g A and g B. u g A + g B c if the good is to be provided.

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When Should a Public Good Be Provided? u Payments must be individually rational; i.e. and

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When Should a Public Good Be Provided? u Payments must be individually rational; i.e. and u Therefore, necessarily and

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When Should a Public Good Be Provided? u And if and then it is Pareto-improving to supply the unit of good

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When Should a Public Good Be Provided? u And if and then it is Pareto-improving to supply the unit of good, so is sufficient for it to be efficient to supply the good.

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Private Provision of a Public Good? u Suppose and. u Then A would supply the good even if B made no contribution. u B then enjoys the good for free; free- riding.

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Private Provision of a Public Good? u Suppose and. u Then neither A nor B will supply the good alone.

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Private Provision of a Public Good? u Suppose and. u Then neither A nor B will supply the good alone. u Yet, if also, then it is Pareto- improving for the good to be supplied.

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Private Provision of a Public Good? u Suppose and. u Then neither A nor B will supply the good alone. u Yet, if also, then it is Pareto- improving for the good to be supplied. u A and B may try to free-ride on each other, causing no good to be supplied.

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Free-Riding u Suppose A and B each have just two actions -- individually supply a public good, or not. u Cost of supply c = $100. u Payoff to A from the good = $80. u Payoff to B from the good = $65.

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Free-Riding u Suppose A and B each have just two actions -- individually supply a public good, or not. u Cost of supply c = $100. u Payoff to A from the good = $80. u Payoff to B from the good = $65. u $80 + $65 > $100, so supplying the good is Pareto-improving.

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Free-Riding Buy Dont Buy Buy Dont Buy Player A Player B

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Free-Riding Buy Dont Buy Buy Dont Buy Player A Player B (Dont Buy, Dont Buy) is the unique NE.

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Free-Riding Buy Dont Buy Buy Dont Buy Player A Player B But (Dont Buy, Dont Buy) is inefficient.

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Free-Riding u Now allow A and B to make contributions to supplying the good. u E.g. A contributes $60 and B contributes $40. u Payoff to A from the good = $40 > $0. u Payoff to B from the good = $25 > $0.

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Free-Riding Contribute Dont Contribute Contribute Dont Contribute Player A Player B

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Free-Riding Contribute Dont Contribute Contribute Dont Contribute Player A Player B Two NE: (Contribute, Contribute) and (Dont Contribute, Dont Contribute).

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Free-Riding u So allowing contributions makes possible supply of a public good when no individual will supply the good alone. u But what contribution scheme is best? u And free-riding can persist even with contributions.

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Variable Public Good Quantities u E.g. how many broadcast TV programs, or how much land to include into a national park.

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Variable Public Good Quantities u E.g. how many broadcast TV programs, or how much land to include into a national park. u c(G) is the production cost of G units of public good. u Two individuals, A and B. u Private consumptions are x A, x B.

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Variable Public Good Quantities u Budget allocations must satisfy

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Variable Public Good Quantities u Budget allocations must satisfy u MRS A & MRS B are A & Bs marg. rates of substitution between the private and public goods. u Pareto efficiency condition for public good supply is

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Variable Public Good Quantities u Pareto efficiency condition for public good supply is u Why?

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Variable Public Good Quantities u Pareto efficiency condition for public good supply is u Why? u The public good is nonrival in consumption, so 1 extra unit of public good is fully consumed by both A and B.

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Variable Public Good Quantities u Suppose u MRS A is As utility-preserving compensation in private good units for a one-unit reduction in public good. u Similarly for B.

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Variable Public Good Quantities u is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit.

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Variable Public Good Quantities u is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit. u Since, making 1 less public good unit releases more private good than the compensation payment requires Pareto- improvement from reduced G.

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Variable Public Good Quantities u Now suppose

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

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Variable Public Good Quantities u Now suppose u is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit. u This payment provides more than 1 more public good unit Pareto- improvement from increased G.

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Variable Public Good Quantities u Hence, necessarily, efficient public good production requires

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Variable Public Good Quantities u Hence, necessarily, efficient public good production requires u Suppose there are n consumers; i = 1,…,n. Then efficient public good production requires

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Efficient Public Good Supply -- the Quasilinear Preferences Case u Two consumers, A and B. u

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Efficient Public Good Supply -- the Quasilinear Preferences Case u Two consumers, A and B. u u Utility-maximization requires

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Efficient Public Good Supply -- the Quasilinear Preferences Case u Two consumers, A and B. u u Utility-maximization requires u is is public good demand/marg. utility curve; i = A,B.

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Efficient Public Good Supply -- the Quasilinear Preferences Case MU A MU B pGpG G

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Efficient Public Good Supply -- the Quasilinear Preferences Case MU A MU B MU A +MU B pGpG G

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Efficient Public Good Supply -- the Quasilinear Preferences Case pGpG MU A MU B MU A +MU B MC(G) G

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Efficient Public Good Supply -- the Quasilinear Preferences Case G pGpG MU A MU B MU A +MU B MC(G) G*

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Efficient Public Good Supply -- the Quasilinear Preferences Case G pGpG MU A MU B MU A +MU B MC(G) G* pG*pG*

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Efficient Public Good Supply -- the Quasilinear Preferences Case G pGpG MU A MU B MU A +MU B MC(G) G* pG*pG*

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Efficient Public Good Supply -- the Quasilinear Preferences Case G pGpG MU A MU B MU A +MU B MC(G) G* pG*pG* Efficient public good supply requires A & B to state truthfully their marginal valuations.

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Free-Riding Revisited u When is free-riding individually rational?

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Free-Riding Revisited u When is free-riding individually rational? u Individuals can contribute only positively to public good supply; nobody can lower the supply level.

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Free-Riding Revisited u When is free-riding individually rational? u Individuals can contribute only positively to public good supply; nobody can lower the supply level. u Individual utility-maximization may require a lower public good level. u Free-riding is rational in such cases.

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Free-Riding Revisited u Given A contributes g A units of public good, Bs problem is subject to

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Free-Riding Revisited G xBxB gAgA Bs budget constraint; slope = -1

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Free-Riding Revisited G xBxB gAgA Bs budget constraint; slope = -1 is not allowed

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Free-Riding Revisited G xBxB gAgA Bs budget constraint; slope = -1 is not allowed

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Free-Riding Revisited G xBxB gAgA Bs budget constraint; slope = -1 is not allowed

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Free-Riding Revisited G xBxB gAgA Bs budget constraint; slope = -1 is not allowed (i.e. free-riding) is best for B

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Demand Revelation u A scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a revelation mechanism. u E.g. the Groves-Clarke taxation scheme. u How does it work?

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Demand Revelation u N individuals; i = 1,…,N. u All have quasi-linear preferences. u v i is individual is true (private) valuation of the public good. u Individual i must provide c i private good units if the public good is supplied.

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Demand Revelation u n i = v i - c i is net value, for i = 1,…,N. u Pareto-improving to supply the public good if

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Demand Revelation u n i = v i - c i is net value, for i = 1,…,N. u Pareto-improving to supply the public good if

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Demand Revelation u If and or and then individual j is pivotal; i.e. changes the supply decision. Schlüsselperson

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Demand Revelation u What loss does a pivotal individual j inflict on others?

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Demand Revelation u What loss does a pivotal individual j inflict on others? u If then is the loss.

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Demand Revelation u What loss does a pivotal individual j inflict on others? u If then is the loss.

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Demand Revelation u For efficiency, a pivotal agent must face the full cost or benefit of her action. u 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.

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Demand Revelation u The GC tax scheme: u Assign a cost c i to each individual. u Each agent states a public good net valuation, s i. u Public good is supplied if otherwise not.

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Demand Revelation u A pivotal person j who changes the outcome from supply to not supply pays a tax of

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Demand Revelation u A pivotal person j who changes the outcome from supply to not supply pays a tax of u A pivotal person j who changes the outcome from not supply to supply pays a tax of

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Demand Revelation u Note: Taxes are not paid to other individuals, but to some other agent outside the market.

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Demand Revelation u Why is the GC tax scheme a revelation mechanism?

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Demand Revelation u Why is the GC tax scheme a revelation mechanism? u An example: 3 persons; A, B and C. u Valuations of the public good are: $40 for A, $50 for B, $110 for C. u Cost of supplying the good is $180.

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Demand Revelation u Why is the GC tax scheme a revelation mechanism? u An example: 3 persons; A, B and C. u Valuations of the public good are: $40 for A, $50 for B, $110 for C. u Cost of supplying the good is $180. u $180 < $40 + $50 + $110 so it is efficient to supply the good.

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Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.

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

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

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

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Demand Revelation u If B and C are truthful, then what net valuation s A should A state? u If s A > -$20, then A makes supply of the public good, and a loss of $20 to him, more likely.

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Demand Revelation u If B and C are truthful, then what net valuation s A should A state? u If s A > -$20, then A makes supply of the public good, and a loss of $20 to him, more likely. u A prevents supply by becoming pivotal, requiring s A + $( ) + $( ) < 0; I.e. A must state s A < -$40.

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Demand Revelation u Then A suffers a GC tax of -$10 + $50 = $40, u As net payoff is - $40 < -$20.

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Demand Revelation u Then A suffers a GC tax of -$10 + $50 = $40, u As net payoff is - $20 - $40 = -$60 < -$20. u A can do no better than state the truth; s A = -$20.

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Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.

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Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u A & Cs net valuations sum to $( ) + $( ) = $30 > 0. u A, B & Cs net valuations sum to u $( ) + $30 = $20 > 0.

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

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Demand Revelation u What net valuation s B should B state?

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Demand Revelation u What net valuation s B should B state? u If s B > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.

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Demand Revelation u What net valuation s B should B state? u If s B > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely. u B prevents supply by becoming pivotal, requiring s B + $( ) + $( ) < 0; I.e. B must state s B < -$30.

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Demand Revelation u Then B suffers a GC tax of -$20 + $50 = $30, u Bs net payoff is - $10 - $30 = -$40 < -$10. u B can do no better than state the truth; s B = -$10.

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Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.

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

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

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Demand Revelation u What net valuation s C should C state?

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Demand Revelation u What net valuation s C should C state? u s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0.

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Demand Revelation u What net valuation s C should C state? u s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0. u s C < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.

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Demand Revelation u What net valuation s C should C state? u s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$( ) - $( ) = $30, for a net payoff of $( ) - $30 = $20 > $0. u s C < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50. u C can do no better than state the truth; s C = $50.

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Grove-Clark Mechanism Telling the truth is a dominant strategy (it is better to tell the truth whatever the others say) Telling the truth is a dominant strategy (it is better to tell the truth whatever the others say)

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The other two say b,c The first player true valuation is a*, but he says a. Assume b+c > 0 1.If a*+b+c > 0 [a*> -(b+c) ] then if he calls a such that a+b+c > 0, he gets a*. If he calls a such that a+b+c < 0 then if he gets -(b+c). But a*> -(b+c) He cannot get anything better than telling the truth a*.

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The other two say b,c The first player true valuation is a*, but he says a. 2.If a*+b+c < 0 [a*< -(b+c) ] then if he calls a such that a+b+c < 0, he gets -(b+c). If he calls a such that a+b+c > 0 then if he gets a*. But a*< -(b+c) He cannot get anything better than telling the truth a*. Assume b+c < 0 and a*+b+c 0 Assume b+c > 0

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Demand Revelation u GC tax scheme implements efficient supply of the public good.

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Demand Revelation u GC tax scheme implements efficient supply of the public good. u But, causes an inefficiency due to taxes removing private good from pivotal individuals.

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