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Variable Public Good Quantities 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 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 x A, x B.

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Variable Public Good Quantities Budget allocations must satisfy

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

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

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Variable Public Good Quantities Pareto efficiency condition for public good supply is Why? 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 Suppose MRS A is A’s utility-preserving compensation in private good units for a one-unit reduction in public good. Similarly for B.

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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.

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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. 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 Now suppose

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Variable Public Good Quantities Now suppose 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 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.

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

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

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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?

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

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

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

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

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

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

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

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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.

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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.

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

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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

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

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

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Demand Revelation Why is the GC tax scheme a revelation 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 a revelation 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 is efficient 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 & 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.

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

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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 to him, 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 to him, more likely. A prevents supply by becoming pivotal, requiring s 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, A’s net payoff is - $20 - $40 = -$60 < -$20.

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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; s A = -$20.

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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. 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.

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

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Demand Revelation What net valuation s B should B state?

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Demand Revelation What net valuation s B should B state? 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 What net valuation s B should B state? If s B > -$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 s B + $(40 - 60) + $(110 - 60) < 0; I.e. B must state s B < -$30.

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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; s B = -$10.

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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. 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.

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

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Demand Revelation What net valuation s C should C state?

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Demand Revelation What net valuation s C should C state? s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.

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Demand Revelation What net valuation s C should C state? s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0. 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 What net valuation s C should C state? s C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0. s C < $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; s C = $50.

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

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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.

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