Chapter Thirty-Five Public Goods
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.
Public Goods -- Examples u Broadcast radio and TV programs. u National defense. u Public highways. u Reductions in air pollution. u National parks.
Reservation Prices u. A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumer’s wealth is u Utility of not having the good is
Reservation Prices u. A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumer’s wealth is u Utility of not having the good is u Utility of paying p for the good is
Reservation Prices u. A consumer’s reservation price for a unit of a good is his maximum willingness-to-pay for it. u Consumer’s 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
Reservation Prices; An Example Consumer’s utility is Utility of not buying a unit of good 2 is Utility of buying one unit of good 2 at price p is
Reservation Prices; An Example Reservation price r is defined by I. e. by
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.
When Should a Public Good Be Provided? u Payments must be individually rational; i. e. and
When Should a Public Good Be Provided? u Payments must be individually rational; i. e. and u Therefore, necessarily and
When Should a Public Good Be Provided? u And if and then it is Pareto-improving to supply the unit of good
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.
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; freeriding.
Private Provision of a Public Good? u Suppose and. u Then neither A nor B will supply the good alone.
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.
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.
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.
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.
Free-Riding Player B Don’t Buy Buy Player A Don’t Buy
Free-Riding Player B Don’t Buy Buy Player A Don’t Buy (Don’t’ Buy, Don’t Buy) is the unique NE.
Free-Riding Player B Don’t Buy Buy Player A Don’t Buy But (Don’t’ Buy, Don’t Buy) is inefficient.
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.
Free-Riding Player B Don’t Contribute Player A Don’t Contribute
Free-Riding Player B Don’t Contribute Player A Don’t Contribute Two NE: (Contribute, Contribute) and (Don’t Contribute, Don’t Contribute).
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.
Variable Public Good Quantities u E. g. how many broadcast TV programs, or how much land to include into a national park.
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.
Variable Public Good Quantities u Budget allocations must satisfy
Variable Public Good Quantities u Budget u MRSA allocations must satisfy & MRSB are A & B’s marg. rates of substitution between the private and public goods. u Pareto efficiency condition for public good supply is
Variable Public Good Quantities u Pareto efficiency condition for public good supply is u Why?
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.
Variable Public Good Quantities u Suppose u MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good. u Similarly for B.
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.
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. u Since , making 1 less public good unit releases more private good than the compensation payment requires Pareto-improvement from reduced G. u
Variable Public Good Quantities u Now suppose
Variable Public Good Quantities u Now u 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 Quantities u Now suppose 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 Paretoimprovement from increased G. u
Variable Public Good Quantities u Hence, necessarily, efficient public good production requires
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
Efficient Public Good Supply -the Quasilinear Preferences Case u Two u consumers, A and B.
Efficient Public Good Supply -the Quasilinear Preferences Case u Two consumers, A and B. u u u Utility-maximization requires
Efficient Public Good Supply -the Quasilinear Preferences Case u Two consumers, A and B. u u u Utility-maximization u requires is i’s public good demand/marg. utility curve; i = A, B.
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUB MUA G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MUA G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MC(G) MUA G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MC(G) MUB MUA G* G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MC(G) MUB p G* MUA G* G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MC(G) MUB p G* MUA G* G
Efficient Public Good Supply -the Quasilinear Preferences Case p. G MUA+MUB MC(G) p G* MUA G* G Efficient public good supply requires A & B to state truthfully their marginal valuations.
Free-Riding Revisited u When is free-riding individually rational?
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.
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.
Free-Riding Revisited u Given A contributes g. A units of public good, B’s problem is subject to
Free-Riding Revisited G B’s budget constraint; slope = -1 g. A x. B
Free-Riding Revisited G B’s budget constraint; slope = -1 g. A is not allowed x. B
Free-Riding Revisited G B’s budget constraint; slope = -1 g. A is not allowed x. B
Free-Riding Revisited G B’s budget constraint; slope = -1 g. A is not allowed x. B
Free-Riding Revisited G B’s budget constraint; slope = -1 g. A (i. e. free-riding) is best for B is not allowed x. B
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?
Demand Revelation u. N individuals; i = 1, …, N. u All have quasi-linear preferences. u vi is individual i’s true (private) valuation of the public good. u Individual i must provide ci private good units if the public good is supplied.
Demand Revelation u ni = vi - ci is net value, for i = 1, …, N. u Pareto-improving to supply the public good if
Demand Revelation u ni = vi - ci is net value, for i = 1, …, N. u Pareto-improving to supply the public good if
Demand Revelation u If or and then individual j is pivotal; i. e. changes the supply decision.
Demand Revelation u What loss does a pivotal individual j inflict on others?
Demand Revelation u What loss does a pivotal individual j inflict on others? u If then is the loss.
Demand Revelation u What loss does a pivotal individual j inflict on others? u If then is the loss.
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.
Demand Revelation u The GC tax scheme: u Assign a cost ci to each individual. u Each agent states a public good net valuation, si. u Public good is supplied if otherwise not.
Demand Revelation u. A pivotal person j who changes the outcome from supply to not supply pays a tax of
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
Demand Revelation u Note: Taxes are not paid to other individuals, but to some other agent outside the market.
Demand Revelation u Why is the GC tax scheme a revelation mechanism?
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.
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.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u B & C’s net valuations sum to $(50 - 60) + $(110 - 60) = $40 > 0. u A, B & C’s net valuations sum to u $(40 - 60) + $40 = $20 > 0.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u B & C’s net valuations sum to $(50 - 60) + $(110 - 60) = $40 > 0. u A, B & C’s net valuations sum to u $(40 - 60) + $40 = $20 > 0. u So A is not pivotal.
Demand Revelation u If B and C are truthful, then what net valuation s. A should A state?
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.
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 + $(50 - 60) + $(110 - 60) < 0; I. e. A must state s. A < -$40.
Demand Revelation u Then A suffers a GC tax of -$10 + $50 = $40, u A’s net payoff is - $20 - $40 = -$60 < -$20.
Demand Revelation u Then A suffers a GC tax of -$10 + $50 = $40, u A’s net payoff is - $20 - $40 = -$60 < -$20. u A can do no better than state the truth; s. A = -$20.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u A & C’s net valuations sum to $(40 - 60) + $(110 - 60) = $30 > 0. u A, B & C’s net valuations sum to u $(50 - 60) + $30 = $20 > 0.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u A & C’s net valuations sum to $(40 - 60) + $(110 - 60) = $30 > 0. u A, B & C’s net valuations sum to u $(50 - 60) + $30 = $20 > 0. u So B is not pivotal.
Demand Revelation u What net valuation s. B should B state?
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.
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 + $(40 - 60) + $(110 - 60) < 0; I. e. B must state s. B < -$30.
Demand Revelation u Then B suffers a GC tax of -$20 + $50 = $30, u B’s net payoff is - $10 - $30 = -$40 < -$10. u B can do no better than state the truth; s. B = -$10.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u A & B’s net valuations sum to $(40 - 60) + $(50 - 60) = -$30 < 0. u A, B & C’s net valuations sum to u $(110 - 60) - $30 = $20 > 0.
Demand Revelation u Assign c 1 = $60, c 2 = $60, c 3 = $60. u A & B’s net valuations sum to $(40 - 60) + $(50 - 60) = -$30 < 0. u A, B & C’s net valuations sum to u $(110 - 60) - $30 = $20 > 0. u So C is pivotal.
Demand Revelation u What net valuation s. C should C state?
Demand Revelation u What net valuation s. C should C state? u s. C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $30 = $20 > $0.
Demand Revelation u What net valuation s. C should C state? u s. C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $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.
Demand Revelation u What net valuation s. C should C state? u s. C > $50 changes nothing. C stays pivotal and must pay a GC tax of -$(40 - 60) - $(50 - 60) = $30, for a net payoff of $(110 - 60) - $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.
Demand Revelation u GC tax scheme implements efficient supply of the public good.
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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