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Chapter 36 Public Goods Chapter 36 Public Goods

Public Goods -- Definition n A good is purely public if it is both Public Goods -- Definition n A good is purely public if it is both nonexcludable and nonrival in consumption. ¨ Nonexcludable -- all consumers can consume the good. ¨ Nonrival -- one consumer’s consumption doesn’t diminish another’s.

Public Goods -- Examples Broadcast radio and TV programs. n National defense. n Public Public Goods -- Examples Broadcast radio and TV programs. n National defense. n Public highways (nearly). n Clean air. n National parks. n A good climate. n

We will need to recall from chapter 14 that: n A consumer’s reservation price We will need to recall from chapter 14 that: n A consumer’s reservation price for a unit of a good is the maximum he/she is willing to pay for it. n If: ¨ Consumer’s ¨ Utility wealth is w when not having the good is U(w, 0) ¨ Utility of paying p and having one unit of the good is U(w-p, 1) n Then the reservation price r is defined by

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

n Call the utility functions UA and UB and assume that A’s income is n Call the utility functions UA and UB and assume that A’s income is w. A and B’s is w. B. n Payments are individually rational if n and n which implies that g. A ≤r. A and g. B ≤ r. B

n And if and then it is Pareto-improving to supply the unit of good n And if and then it is Pareto-improving to supply the unit of good n (It is Pareto-improving only if one inequality holds with < and the other at least with ≤) n If c ≤ r. A + r. B it must be pareto-efficient to supply the good.

Free-riding Suppose c < r. A and 0 < r. B. n Then A Free-riding Suppose c < r. A and 0 < r. B. n Then A would supply the good even if B made no contribution. n Then B would enjoy the good for free; be free-riding. n

Private Provision of a Public Good? n n Suppose c < r. A and Private Provision of a Public Good? n n Suppose c < r. A and c < r. B. Neither A nor B will supply the good alone. Yet, if c < r. A + r. B both could be better off if the good is supplied with a payment scheme where c < g. A + g. B ; g. A < r. A and g. B < r. B But A and B may try to free-ride on each other, causing no good to be supplied.

Free-Riding - an example n n n Suppose A and B each have just Free-Riding - an example n n n 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 it is possible to supply the good and make both better off.

Player B Buy Don’t Buy Player A Don’t Buy (Don’t’ Buy, Don’t Buy) is Player B Buy Don’t Buy Player A Don’t Buy (Don’t’ Buy, Don’t Buy) is the unique NE

But (Don’t buy, Don’t buy) is not efficient. n Why? n If each pays But (Don’t buy, Don’t buy) is not efficient. n Why? n If each pays part of the cost: n For example, A contributes 60 and B contributes 40. n Payoff to A from the good = 40 > 0. n Payoff to B from the good = 25 > 0. n Any division of 100 such that g. A≤ 80 and g. B ≤ 65 will do. n

With cost-sharing there are two NE. But even if the good is supplied, there With cost-sharing there are two NE. But even if the good is supplied, there can still be some free-riding. Player B Don’t Contribute Player A Don’t Contribute

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

n Budget allocations must satisfy MRSA & MRSB are A & B’s marg. rates n Budget allocations must satisfy MRSA & MRSB are A & B’s marg. rates of substitution between the private and public goods. n Pareto efficiency condition for public good supply is n

n n n MRSA is A’s utility-preserving compensation in private good units for a n n n MRSA is A’s utility-preserving compensation in private good units for a one-unit reduction in public good. Similarly for B. is ¨ the total payment to A & B of private good that preserves both utilities if G is lowered by 1 unit. ¨ the total amount of private good that A and B together are willing to give up to have G increased by 1 unit

, making 1 unit less of the public good releases more private good than , making 1 unit less of the public good releases more private good than the compensation payment requires Paretoimprovement from reduced G. n If the payment A and B are willing to make for 1 unit of public good provides more than 1 unit Paretoimprovement from increased G. n

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

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

n Each agent decides how much to contribute based on how much everybody else n Each agent decides how much to contribute based on how much everybody else is expected to contribute. ¨ For n example: Given A that contributes g. A units of public good, B’s problem is to choose x. B and g. B so as to maximise UB(x. B, g. A+ g. B) subject to x. B + g. B = w. B and g. B≥ 0

G B’s budget constraint; slope = -1 B’s endowment if A pays g. A G B’s budget constraint; slope = -1 B’s endowment if A pays g. A for the public good g. A w. B x. B

G B’s budget constraint; slope = -1 g. A g. B = 0 (free-riding) G B’s budget constraint; slope = -1 g. A g. B = 0 (free-riding) is best for B g. B<0 is not possible x. B

Demand Revelation A scheme that makes it rational for individuals to reveal truthfully their Demand Revelation A scheme that makes it rational for individuals to reveal truthfully their private valuations of a public good is a revelation mechanism. n For example, the Groves-Clarke taxation scheme n

Assume: n n n N individuals; i = 1, …, N. All have quasi-linear Assume: n n n 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. ni = vi - ci is net value, for i = 1, …, N. Can be Pareto-improving to supply the public good if

n ni = vi - ci is net value, for i = 1, …, n ni = vi - ci is net value, for i = 1, …, N. n If or and then individual j is pivotal; i. e. changes the supply decision.

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

The GC tax scheme: n Assign a cost ci to each individual. n Each The GC tax scheme: n Assign a cost ci to each individual. n Each agent states a public good net valuation, si. n Public good is supplied if otherwise not. n

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

n GC tax scheme implements efficient supply of the public good. n Note: Taxes n GC tax scheme implements efficient supply of the public good. n Note: Taxes are not paid to other individuals, but to some other agent outside the market.

An example: 3 persons; A, B and C. n Valuations of the public good An example: 3 persons; A, B and C. n Valuations of the public good are: 40 for A, 50 for B, 110 for C. n Cost of supplying the good is 180. n 180 < 40 + 50 + 110 so it is efficient to supply the good. n Assign c 1 = 60, c 2 = 60, c 3 = 60. n

B & C’s net valuations sum to (50 - 60) + (110 - 60) B & C’s net valuations sum to (50 - 60) + (110 - 60) = 40 > 0. n A, B & C’s net valuations sum to n (40 - 60) + 40 = 20 > 0. n So A is not pivotal. n A’s true net value is 40 – 60 = -20 n If s. A > -20, then A makes supply of the public good, and a loss of 20 to him, more likely. n

n n If B and C are truthful, then what net valuation s. A n n 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. If s. A < -20 but not enough to make A pivotal his loss is still 20. A prevents supply by becoming pivotal, only if s. A + (50 - 60) + (110 - 60) < 0; To be pivotal A must state s. A < -40.

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

Exercise: n Use the same method to show that n ¨B is not pivotal Exercise: n Use the same method to show that n ¨B is not pivotal ¨ B can do no better than state the truth; s. B = -10. ¨ C is pivotal ¨ C can do no better than state the truth; s. C = 50.