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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. 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.

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variable public good quantities
Variable Public Good Quantities
  • E.g. how many broadcast TV programs, or how much land to include into a national park.
variable public good quantities1
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 xA, xB.
variable public good quantities2
Variable Public Good Quantities
  • Budget allocations must satisfy
variable public good quantities3
Variable Public Good Quantities
  • Budget allocations must satisfy
  • MRSA & MRSB are A & B’s marg. rates of substitution between the private and public goods.
  • Pareto efficiency condition for public good supply is
variable public good quantities4
Variable Public Good Quantities
  • Pareto efficiency condition for public good supply is
  • Why?
variable public good quantities5
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.
variable public good quantities6
Variable Public Good Quantities
  • Suppose
  • MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good.
  • Similarly for B.
variable public good quantities7
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 quantities8
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.
variable public good quantities10
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.
variable public good quantities11
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.
variable public good quantities12
Variable Public Good Quantities
  • Hence, necessarily, efficient public good production requires
variable public good quantities13
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
demand revelation
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?
demand revelation1
Demand Revelation
  • N individuals; i = 1,…,N.
  • All have quasi-linear preferences.
  • vi is individual i’s true (private) valuation of the public good.
  • Individual i must provide ci private good units if the public good is supplied.
demand revelation2
Demand Revelation
  • ni = vi - ci is net value, for i = 1,…,N.
  • Pareto-improving to supply the public good if
demand revelation3
Demand Revelation
  • ni = vi - ci is net value, for i = 1,…,N.
  • Pareto-improving to supply the public good if
demand revelation4
Demand Revelation
  • If andor andthen individual j is pivotal; i.e. changes the supply decision.
demand revelation5
Demand Revelation
  • What loss does a pivotal individual j inflict on others?
demand revelation6
Demand Revelation
  • What loss does a pivotal individual j inflict on others?
  • If then is the loss.
demand revelation7
Demand Revelation
  • What loss does a pivotal individual j inflict on others?
  • If then is the loss.
  • If then is the loss.
demand revelation8
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.
demand revelation9
Demand Revelation
  • The GC tax scheme:
  • Assign a cost ci to each individual.
  • Each agent states a public good net valuation, si.
  • Public good is supplied if otherwise not.
demand revelation10
Demand Revelation
  • A pivotal person j who changes the outcome from supply to not supplypays a tax of
demand revelation11
Demand Revelation
  • A pivotal person j who changes the outcome from supply to not supplypays a tax of
  • A pivotal person j who changes the outcome from not supply to supplypays a tax of
demand revelation12
Demand Revelation
  • Note: Taxes are not paid to other individuals, but to some other agent outside the market.
demand revelation13
Demand Revelation
  • Why is the GC tax scheme a revelation mechanism?
demand revelation14
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.
demand revelation15
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.
demand revelation16
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
demand revelation17
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • B & C’s net valuations sum to$(50 - 60) + $(110 - 60) = $40 > 0.
  • A, B & C’s net valuations sum to
  • $(40 - 60) + $40 = $20 > 0.
demand revelation18
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • B & C’s net valuations sum to$(50 - 60) + $(110 - 60) = $40 > 0.
  • A, B & C’s net valuations sum to
  • $(40 - 60) + $40 = $20 > 0.
  • So A is not pivotal.
demand revelation19
Demand Revelation
  • If B and C are truthful, then what net valuation sA should A state?
demand revelation20
Demand Revelation
  • If 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.
demand revelation21
Demand Revelation
  • If 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;I.e. A must state sA < -$40.
demand revelation22
Demand Revelation
  • Then A suffers a GC tax of -$10 + $50 = $40,
  • A’s net payoff is - $20 - $40 = -$60 < -$20.
demand revelation23
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; sA = -$20.
demand revelation24
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
demand revelation25
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • A & C’s net valuations sum to$(40 - 60) + $(110 - 60) = $30 > 0.
  • A, B & C’s net valuations sum to
  • $(50 - 60) + $30 = $20 > 0.
demand revelation26
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • A & C’s net valuations sum to$(40 - 60) + $(110 - 60) = $30 > 0.
  • A, B & C’s net valuations sum to
  • $(50 - 60) + $30 = $20 > 0.
  • So B is not pivotal.
demand revelation27
Demand Revelation
  • What net valuation sB should B state?
demand revelation28
Demand Revelation
  • What net valuation sB should B state?
  • If sB > -$10, then B makes supply of the public good, and a loss of $10 to him, more likely.
demand revelation29
Demand Revelation
  • What net valuation sB should B state?
  • If sB > -$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 sB + $(40 - 60) + $(110 - 60) < 0;I.e. B must state sB < -$30.
demand revelation30
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; sB = -$10.
demand revelation31
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
demand revelation32
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • A & B’s net valuations sum to$(40 - 60) + $(50 - 60) = -$30 < 0.
  • A, B & C’s net valuations sum to
  • $(110 - 60) - $30 = $20 > 0.
demand revelation33
Demand Revelation
  • Assign c1 = $60, c2 = $60, c3 = $60.
  • A & B’s net valuations sum to$(40 - 60) + $(50 - 60) = -$30 < 0.
  • A, B & C’s net valuations sum to
  • $(110 - 60) - $30 = $20 > 0.
  • So C is pivotal.
demand revelation34
Demand Revelation
  • What net valuation sC should C state?
demand revelation35
Demand Revelation
  • What net valuation sC should C state?
  • sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
demand revelation36
Demand Revelation
  • What net valuation sC should C state?
  • sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
  • sC < $50 makes it less likely that the public good will be supplied, in which case C loses $110 - $60 = $50.
demand revelation37
Demand Revelation
  • What net valuation sC should C state?
  • sC > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
  • sC < $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; sC = $50.
demand revelation38
Demand Revelation
  • GC tax scheme implements efficient supply of the public good.
demand revelation39
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.