Microeconomics (Public Goods Ch 36 (Varian))

Lectures 26 & 27

Apr 24 & 27, 2017 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. Public Goods -- Examples

Broadcast radio and TV programs. National defense. Public highways. Reductions in air pollution. National parks. 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). 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 Prices

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 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 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 gA and gB.

gA + gB c if the good is to be provided. When Should a Public Good Be Provided?

Payments must be individually rational; i.e. UA (wA ,0) UA (wA gA ,1) and UB(wB,0) UB(wB gB,1). When Should a Public Good Be Provided?

Payments must be individually rational; i.e. UA (wA ,0) UA (wA gA ,1) and UB(wB,0) UB(wB gB,1). Therefore, necessarily gA rA and gB rB. When Should a Public Good Be Provided?

And if UA (wA ,0) UA (wA gA ,1) and UB(wB,0) UB(wB gB,1) then it is Pareto-improving to supply the unit of good When Should a Public Good Be Provided?

And if UA (wA ,0) UA (wA gA ,1) and 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. Private Provision of a Public Good?

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

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

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

Suppose r A c and r B c . Then neither A nor B will supply the good alone. Yet, if r A r B c also, then it is Pareto- improving for the good to be supplied. A and B may try to free-ride on each other, causing no good to be supplied. 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. 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. Free-Riding

Player B Don’t Buy Buy

Buy -$20, -$35 -$20, $65 Player A Don’t Buy $100, -$35 $0, $0 Free-Riding

Player B Don’t Buy Buy

Buy -$20, -$35 -$20, $65 Player A Don’t Buy $100, -$35 $0, $0

(Don’t’ Buy, Don’t Buy) is the unique NE. Free-Riding

Player B Don’t Buy Buy

Buy -$20, -$35 -$20, $65 Player A Don’t Buy $100, -$35 $0, $0

But (Don’t’ Buy, Don’t Buy) is inefficient. 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. Free-Riding

Player B Don’t Contribute Contribute

Contribute $20, $25 -$60, $0 Player A Don’t Contribute $0, -$40 $0, $0 Free-Riding

Player B Don’t Contribute Contribute

Contribute $20, $25 -$60, $0 Player A Don’t Contribute $0, -$40 $0, $0

Two NE: (Contribute, Contribute) and (Don’t Contribute, Don’t Contribute). 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. 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. Two individuals, A and B.

Private consumptions are xA, xB. Variable Public Good Quantities

Budget allocations must satisfy xA xB c(G) wA wB. Variable Public Good Quantities

Budget allocations must satisfy xA xB c(G) wA wB.

MRSA & MRSB 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). Variable Public Good Quantities

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

Suppose MRSA MRSB MC(G).

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 Quantities

MRSA MRSB 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

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

Now suppose MRSA MRSB MC(G). 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. 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 Pareto- improvement from increased G. Variable Public Good Quantities

Hence, necessarily, efficient public good production requires MRSA MRSB MC(G). 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). i1 Efficient Public Good Supply -- the Quasilinear Preferences Case

Two consumers, A and B. Ui (xi ,G) xi fi (G); i A,B. 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 pG MRSi fi(G) pG ; i A,B. px 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 pG MRSi fi(G) pG ; i A,B. px

pG fi(G) is i’s public good demand/marg. utility curve; i = A,B. Efficient Public Good Supply -- the Quasilinear Preferences Case

MUB

MUA G Efficient Public Good Supply -- the Quasilinear Preferences Case

MUA+MUB

MUB

MUA G Efficient Public Good Supply -- the Quasilinear Preferences Case

MUA+MUB

MUB MC(G)

MUA G Efficient Public Good Supply -- the Quasilinear Preferences Case

MUA+MUB

MUB MC(G)

MUA G* G Efficient Public Good Supply -- the Quasilinear Preferences Case

MUA+MUB

MUB MC(G)

pG*

MUA G* G Efficient Public Good Supply -- the Quasilinear Preferences Case

pG * pG MUA (G*) MUB (G*) MUA+MUB

MUB MC(G)

pG*

MUA G* G Efficient Public Good Supply -- the Quasilinear Preferences Case

pG * pG MUA (G*) MUB (G*) MUA+MUB

MUB MC(G) pG*

MUA G* G Efficient public good supply requires A & B to state truthfully their marginal valuations. Free-Riding Revisited

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

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

Given A contributes gA units of public good, B’s problem is max UB(xB, gA gB ) xB ,gB subject to xB gB wB, gB 0. Free-Riding Revisited

B’s budget constraint; slope = -1

xB Free-Riding Revisited

B’s budget constraint; slope = -1 gB 0 gA gB 0 is not allowed

xB Free-Riding Revisited

B’s budget constraint; slope = -1 gB 0 gA gB 0 is not allowed

xB Free-Riding Revisited

B’s budget constraint; slope = -1 gB 0 gA gB 0 is not allowed

xB Free-Riding Revisited

B’s budget constraint; slope = -1 gB 0 gB 0 (i.e. free-riding) is best for B gA gB 0 is not allowed

xB 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 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 Revelation

ni = vi - ci is net value, for i = 1,…,N. Pareto-improving to supply the public good if N N vi ci i1 i1 Demand Revelation

ni = vi - ci is net value, for i = 1,…,N. Pareto-improving to supply the public good if N N N vi ci ni 0. i1 i1 i1 Demand Revelation

N N If 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. Demand Revelation

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

The GC tax scheme:

Assign a cost ci to each individual. Each agent states a public good net valuation, s . i N Public good is supplied if si 0; otherwise not. i1 Demand Revelation

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

A pivotal person j who changes the outcome from supply to not supply N pays a tax of si . i j

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

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

Why is the GC tax scheme a revelation mechanism? 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 Revelation

Assign c1 = $60, c2 = $60, c3 = $60. 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 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 Revelation

If B and C are truthful, then what net

valuation sA should A state? 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 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 Revelation

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

Assign c1 = $60, c2 = $60, c3 = $60. 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 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 Revelation

What net valuation sB should B state? 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 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 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 Revelation

Assign c1 = $60, c2 = $60, c3 = $60. 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 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 Revelation

What net valuation sC should C state? 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 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 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 Revelation

GC tax scheme implements efficient supply of the public good. 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. Microeconomics (Public Goods Ch 36 (Varian))

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