Microeconomics (Public Goods Ch 36 (Varian))

Transcription

1 Microeconomics (Public Goods Ch 36 (Varian)) Microeconomics (Public Goods Ch 36 (Varian)) Lectures 26 & 27 Apr 24 & 27, 2017

2 Public Goods -- Definition 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.

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

4 Reservation Prices A consumer s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer s wealth is w. Utility of not having the good is U( w, 0).

5 Reservation Prices A consumer s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer s wealth is w. Utility of not having the good is U( w, 0). Utility of paying p for the good is U( w p, 1).

6 Reservation Prices A consumer s reservation price for a unit of a good is his maximum willingness-to-pay for it. Consumer s wealth is w. Utility of not having the good is U( w, 0). Utility of paying p for the good is U( w p, 1). Reservation price r is defined by U( w, 0) U( w r, 1).

7 Reservation Prices; An Example Consumer s utility is U( x1, x2) x1( x2 1). Utility of not buying a unit of good 2 is w w V ( w, 0) ( 0 1). p1 p1 Utility of buying one unit of good 2 at price p is w p 2( w p) V ( w p, 1) ( 1 1). p1 p1

8 Reservation Prices; An Example Reservation price r is defined by V ( w, 0) V ( w r, 1) I.e. by w 2( w r) w r. p1 p1 2

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

10 When Should a Public Good Be Provided? Payments must be individually rational; i.e. and U A( w A, 0) U A( w A g A, 1) UB( wb, 0) UB( wb gb, 1).

11 When Should a Public Good Be Provided? Payments must be individually rational; i.e. and UA( wa, 0) UA( wa ga, 1) UB( wb, 0) UB( wb gb, 1). Therefore, necessarily ga ra and gb rb.

12 When Should a Public Good Be Provided? And if and UA( wa, 0) UA( wa ga, 1) UB( wb, 0) UB( wb gb, 1) then it is Pareto-improving to supply the unit of good

13 When Should a Public Good Be Provided? And if and UA( wa, 0) UA( wa ga, 1) UB( wb, 0) UB( wb gb, 1) then it is Pareto-improving to supply the unit of good, so ra rb c is sufficient for it to be efficient to supply the good.

14 Private Provision of a Public Good? Suppose ra c and r B c. Then A would supply the good even if B made no contribution. B then enjoys the good for free; freeriding.

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

16 Private Provision of a Public Good? Suppose ra c and r B c. Then neither A nor B will supply the good alone. Yet, if ra rb c also, then it is Paretoimproving for the good to be supplied.

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

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

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

20 Free-Riding Player B Don t Buy Buy Buy Player A Don t Buy -$20, -$35 -$20, $65 $100, -$35 $0, $0

21 Free-Riding Player B Don t Buy Buy Buy Player A Don t Buy -$20, -$35 -$20, $65 $100, -$35 $0, $0 (Don t Buy, Don t Buy) is the unique NE.

22 Free-Riding Player B Don t Buy Buy Buy Player A Don t Buy -$20, -$35 -$20, $65 $100, -$35 $0, $0 But (Don t Buy, Don t Buy) is inefficient.

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

24 Free-Riding Player B Don t Contribute Contribute Contribute Player A Don t Contribute $20, $25 -$60, $0 $0, -$40 $0, $0

25 Free-Riding Player B Don t Contribute Contribute Contribute Player A Don t Contribute $20, $25 -$60, $0 $0, -$40 $0, $0 Two NE: (Contribute, Contribute) and (Don t Contribute, Don t Contribute).

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

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

28 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.

29 Variable Public Good Quantities Budget allocations must satisfy xa xb c( G) wa wb.

30 Variable Public Good Quantities Budget allocations must satisfy xa xb c( G) wa wb. 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 MRSA MRSB MC( G).

31 Variable Public Good Quantities Pareto efficiency condition for public good supply is MRSA MRSB MC( G). Why?

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

33 Variable Public Good Quantities Suppose MRSA MRSB MC( G). MRS A is A s utility-preserving compensation in private good units for a one-unit reduction in public good. Similarly for B.

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

35 Variable Public Good Quantities MRSA MRSB is the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit. Since MRSA MRSB MC( G), making 1 less public good unit releases more private good than the compensation payment requires Paretoimprovement from reduced G.

36 Variable Public Good Quantities Now suppose MRSA MRSB MC( G).

37 Variable Public Good Quantities Now suppose MRSA MRSB MC( G). MRSA MRSB is the total payment by A & B of private good that preserves both utilities if G is raised by 1 unit.

38 Variable Public Good Quantities Now suppose MRSA MRSB MC( G). MRSA MRSB 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 Paretoimprovement from increased G.

39 Variable Public Good Quantities Hence, necessarily, efficient public good production requires MRSA MRSB MC( G).

40 Variable Public Good Quantities Hence, necessarily, efficient public good production requires MRSA MRSB MC( G). Suppose there are n consumers; i = 1,,n. Then efficient public good production requires n MRSi MC( G). i 1

41 Efficient Public Good Supply -- the Quasilinear Preferences Case Two consumers, A and B. Ui ( xi, G) xi fi ( G); i A, B.

42 Efficient Public Good Supply -- the Quasilinear Preferences Case Two consumers, A and B. Ui ( xi, G) xi fi ( G); i A, B. MRSi fi ( G); i A, B. Utility-maximization requires p MRSi G fi ( G) pg ; i A, B. px

43 Efficient Public Good Supply -- the Quasilinear Preferences Case Two consumers, A and B. Ui ( xi, G) xi fi ( G); i A, B. MRSi fi ( G); i A, B. Utility-maximization requires p MRSi G fi ( G) pg ; i A, B. px pg fi ( G) is i s public good demand/marg. utility curve; i = A,B.

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

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

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

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

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

49 Efficient Public Good Supply -- the Quasilinear Preferences Case p G MU A +MU B pg * MUA( G*) MUB( G*) p G * MU B MC(G) G* MU A G

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

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

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

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

54 Free-Riding Revisited Given A contributes g A units of public good, B s problem is max UB( xb, ga gb ) x, g B B subject to xb gb wb, gb 0.

55 Free-Riding Revisited G B s budget constraint; slope = -1 g A x B

56 Free-Riding Revisited G B s budget constraint; slope = -1 g B 0 g A g B 0 is not allowed x B

57 Free-Riding Revisited G B s budget constraint; slope = -1 g B 0 g A g B 0 is not allowed x B

58 Free-Riding Revisited G B s budget constraint; slope = -1 g B 0 g A g B 0 is not allowed x B

59 Free-Riding Revisited G g A B s budget constraint; slope = -1 g B 0 g B 0 (i.e. free-riding) is best for B g B 0 is not allowed x B

60 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?

61 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.

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

63 Demand Revelation n i = v i - c i is net value, for i = 1,,N. Pareto-improving to supply the public good if N N N vi ci ni 0. i 1 i 1 i 1

64 Demand Revelation If N N n i 0 and ni nj 0 i j i j N N or n i 0 and ni nj 0 i j i j then individual j is pivotal; i.e. changes the supply decision.

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

66 Demand Revelation What loss does a pivotal individual j inflict on others? N N If n i 0, then n i 0 is the loss. i j i j

67 Demand Revelation What loss does a pivotal individual j inflict on others? N N If n i 0, then n i 0 is the loss. i j i j N N If n i 0, then n i 0 is the loss. i j i j

68 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.

69 Demand Revelation The GC tax scheme: Assign a cost c i to each individual. Each agent states a public good net valuation, s i. N Public good is supplied if s i 0; otherwise not. i 1

70 Demand Revelation A pivotal person j who changes the outcome from supply to not supply pays a tax of N s i. i j

71 Demand Revelation A pivotal person j who changes the outcome from supply to not supply pays a tax of N s i. i j A pivotal person j who changes the outcome from not supply to supply pays a tax of N s i. i j

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

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

74 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.

75 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.

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

77 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.

78 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.

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

80 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.

81 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.

82 Demand Revelation Then A suffers a GC tax of -$10 + $50 = $40, A s net payoff is - $20 - $40 = -$60 < -$20.

83 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.

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

85 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.

86 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.

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

88 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.

89 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.

90 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.

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

92 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.

93 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.

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

95 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.

96 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.

97 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.

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

99 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.

100 Microeconomics (Public Goods Ch 36 (Varian)) Thank You