PUBLIC GOODS Snyder and Nicholson, Copyright ©2008 by Thomson South-Western. All rights reserved.

PUBLIC GOODS

Snyder and Nicholson, Copyright ©2008 by Thomson South-Western. All rights reserved.

Public Goods• Public goods are nonrival

– the use of the good does not prevent others from using it e.g. knowledge

• Pure Public goods are nonexclusive– once they are produced, they provide

benefits to an entire group– it is impossible to restrict these benefits to

the specific groups of individuals who pay for them

Attributes of Public Goods• A good is nonrival if consumption of

additional units of the good involves zero social marginal costs of production

Attributes of Public Goods• A good is exclusive if it is relatively easy

to exclude individuals from benefiting from the good once it is produced

• A good is nonexclusive if it is impossible, or very costly, to exclude individuals from benefiting from the good

Attributes of Public Goods

Exclusive

Yes No

Yes Hot dogs,

cars, houses

Fishing grounds, clean air

Rival

No Bridges,

swimming pools

National defense, mosquito control

• Some examples of these types of goods include:

Public Good• A good is a pure public good if, once

produced, no one can be excluded from benefiting from its availability and if the good is nonrival -- the marginal cost of an additional consumer is zero

Public Goods andResource Allocation

• We will use a simple general equilibrium model with two individuals (A and B)

• There are only two goods– good y is an ordinary private good

• each person begins with an allocation

(yA and yB)

– good x is a public good that is produced using y

x = f(ysA + ys

B)


• Resulting utilities for these individuals areUA[x,(yA - ys

A)]

UB[x,(yB - ysB)]

• The level of x enters identically into each person’s utility curve– it is nonexclusive and nonrival

• each person’s consumption is unrelated to what he contributes to production

• each consumes the total amount produced


• The necessary conditions for efficient resource allocation consist of choosing the levels of ys

A and ysB that maximize one

person’s (A’s) utility for any given level of the other’s (B’s) utility

• The Lagrangian expression is

L = UA(x, yA - ysA) + [UB(x, yB - ys

B) - K]


• The first-order conditions for a maximum are

L/ysA = U1

Af’ - U2A + U1

Bf’ = 0

L/ysB = U1

Af’ - U2B + U1

Bf’ = 0

• Comparing the two equations, we find

U2B = U2

A


• We can now derive the optimality condition for the production of x

• From the initial first-order condition we know that

U1A/U2

A + U1B/U2

B = 1/f’

MRSA + MRSB = 1/f’

• The MRS must reflect all consumers because all will get the same benefits

Failure of aCompetitive Market

• Production of x and y in competitive markets will fail to achieve this allocation– with perfectly competitive prices px and py,

each individual will equate his MRS to px/py

– the producer will also set 1/f’ equal to px/py to maximize profits

– the price ratio px/py will be too low• it would provide too little incentive to produce x

Failure of aCompetitive Market

• For public goods, the value of producing one more unit is the sum of each consumer’s valuation of that output– individual demand curves should be added

vertically rather than horizontally

• Thus, the usual market demand curve will not reflect the full marginal valuation

Inefficiency of aNash Equilibrium

• Suppose that individual A is thinking about contributing ys

A of his initial yA endowment to the production of x

• The utility maximization problem for A is then

choose ysA to maximize UA[f(ys

A + ysB),yA - ys

A]

Inefficiency of aNash Equilibrium

• The first-order condition for a maximum is

U1Af’ - U2

A = 0

U1A/U2

A = MRSA = 1/f’

• Because a similar argument can be applied to B, the efficiency condition will fail to be achieved– each person considers only his own benefit

The Roommates’ Dilemma

• Suppose two roommates with identical preferences derive utility from the number of paintings hung on their walls (x) and the number of chocolate bars they eat (y) with a utility function of

Ui(x,yi) = x1/3yi2/3 (for i=1,2)

• Assume each roommate has $300 to spend and that px = $100 and py = $0.20

The Roommates’ Dilemma• We know from our earlier analysis of

Cobb-Douglas utility functions that if each individual lived alone, he would spend 1/3 of his income on paintings (x = 1) and 2/3 on chocolate bars (y = 1,000)

• When the roommates live together, each must consider what the other will do– if each assumed the other would buy

paintings, x = 0 and utility = 0


• If person 1 believes that person 2 will not buy any paintings, he could choose to purchase one and receive utility of

U1(x,y1) = 11/3(1,000)2/3 = 100

while person 2’s utility will be

U2(x,y2) = 11/3(1,500)2/3 = 131

• Person 2 has gained from his free-riding position


• We can show that this solution is inefficient by calculating each person’s MRS

x

y

yU

xUMRS i

ii

ii 2/

/

• At the allocations described,MRS1 = 1,000/2 = 500

MRS2 = 1,500/2 = 750


• Since MRS1 + MRS2 = 1,250, the roommates would be willing to sacrifice 1,250 chocolate bars to have one additional painting– an additional painting would only cost them

500 chocolate bars– too few paintings are bought


• To calculate the efficient level of x, we must set the sum of each person’s MRS equal to the price ratio

20.0

100

2222121

21

y

x

p

p

x

yy

x

y

x

yMRSMRS

• This means that

y1 + y2 = 1,000x


• Substituting into the sum of budget constraints, we get

0.20(y1 + y2) + 100x = 600

x = 2

y1 + y2 = 2,000

• The allocation of the cost of the paintings depends on how each roommate plays the strategic financing game


• If each person buy 1 painting

Person 1’s utility is

U1(x,y1) = 21/3(1,000)2/3 126

Person 2’s utility is

U2(x,y2) = 21/3(1,000)2/3 126


126,126 100,131

131,100 0,0

Person 2

Buy Not Buy

Not Buy

Person 1Buy

Lindahl Pricing ofPublic Goods

• Swedish economist E. Lindahl suggested that individuals might be willing to be taxed for public goods if they knew that others were being taxed– Lindahl assumed that each individual would

be presented by the government with the proportion of a public good’s cost he was expected to pay and then reply with the level of public good he would prefer


• Suppose that individual A would be quoted a specific percentage (A) and asked the level of a public good (x) he would want given the knowledge that this fraction of total cost would have to be paid

• The person would choose the level of x which maximizes

utility = UA[x,yA*- Af -1(x)]


• The first-order condition is given by

U1A - AU2

B(1/f’)=0

MRSA = A/f’

• Faced by the same choice, individual B would opt for the level of x which satisfies

MRSB = B/f’


• An equilibrium would occur when A+B = 1– the level of public goods expenditure

favored by the two individuals precisely generates enough tax contributions to pay for it

MRSA + MRSB = (A + B)/f’ = 1/f’

Shortcomings of theLindahl Solution

• The incentive to be a free rider is very strong– this makes it difficult to envision how the

information necessary to compute equilibrium Lindahl shares might be computed

• individuals have a clear incentive to understate their true preferences

Important Points to Note:• Externalities may cause a

misallocation of resources because of a divergence between private and social marginal cost– traditional solutions to this divergence

includes mergers among the affected parties and adoption of suitable Pigouvian taxes or subsidies

Voting• Voting is used as a social decision

process in many institutions– direct voting is used in many cases from

statewide referenda to smaller groups and clubs

– in other cases, societies have found it more convenient to use a representative form of government

Majority Rule• Throughout our discussion of voting, we

will assume that decisions will be made by majority rule

The Paradox of Voting• In the 1780s, social theorist M. de

Condorcet noted that majority rule voting systems may not arrive at an equilibrium– instead, they may cycle among alternative

options

The Paradox of Voting

• Suppose there are three voters (Smith, Jones, and Fudd) choosing among three policy options– we can assume that these policy options

represent three levels of spending on a particular public good [(A) low, (B) medium, and (C) high]

– Condorcet’s paradox would arise even without this ordering


Smith Jones Fudd

A B C

B C A

C A B

• Preferences among the three policy options for the three voters are:


• Consider a vote between A and B– A would win

• In a vote between A and C– C would win

• In a vote between B and C– B would win

• No equilibrium will ever be reached

Single-Peaked Preferences• Equilibrium voting outcomes always

occur in cases where the issue being voted upon is one-dimensional and where voter preferences are “single-peaked”

Single-Peaked Preferences

Quantity ofpublic good

Utility

A B C

Smith

We can show each voters preferences in terms of utility levels

Jones

Fudd

For Smith and Jones, preferences are single-peaked

Fudd’s preferences have two local maxima

Single-Peaked Preferences

Quantity ofpublic good

Utility

A B C

Smith

Jones

Option B will be chosen because it will defeat both A and C by votes 2 to 1

If Fudd had alternative preferences with a single peak, there would be no paradox

Fudd

The Median Voter Theorem• With the altered preferences of Fudd, B

will be chosen because it is the preferred choice of the median voter (Jones)– Jones’s preferences are between the

preferences of Smith and the revised preferences of Fudd

The Median Voter Theorem• If choices are unidimensional and

preferences are single-peaked, majority rule will result in the selection of the project that is most favored by the median voter– that voter’s preferences will determine

what public choices are made

A Simple Political Model• Suppose a community is characterized

by a large number of voters (n) each with income of yi

• The utility of each voter depends on his consumption of a private good (ci) and of a public good (g) according to

utility of person i = Ui = ci + f(g)

where fg > 0 and fgg < 0

A Simple Political Model

• Each voter must pay taxes to finance g• Taxes are proportional to income and

are imposed at a rate of t• Each person’s budget constraint is

ci = (1-t)yi

• The government also faces a budget constraint

An

ii tnytyg

1

A Simple Political Model• Given these constraints, the utility

function of individual i is

Ui(g) = [yA - (g/n)]yi /yA + f(g)

• Utility maximization occurs when

dUi /dg = -yi /(nyA) + fg(g) = 0

g = fg-1[yi /(nyA)]

• Desired spending on g is inversely related to income

A Simple Political Model

• If G is determined through majority rule, its level will be that level favored by the median voter– since voters’ preferences are determined

solely by income, g will be set at the level preferred by the voter with the median level of income (ym)

g* = fg-1[ym/(nyA)] = fg

-1[(1/n)(ym/yA)]

A Simple Political Model• Under a utilitarian social welfare

criterion, g would be chosen so as to maximize the sum of utilities:

gnfgnygfyy

ng

yUSW AAiA

n

i

1

• The optimal choice for g then is

g* = fg-1(1/n) = fg

-1[(1/n)(yA/yA)]

– the level of g favored by the voter with average income

Voting for Redistributive Taxation

• Suppose voters are considering a lump-sum transfer to be paid to every person and financed through proportional taxation

• If we denote the per-person transfer b, each individual’s utility is now given by

Ui = ci + b


• The government’s budget constraint is

nb = tnyA

b = tyA

• For a voter with yi > yA, utility is maximized by choosing b = 0

• Any voter with yi < yA will choose t = 1 and b = yA

– would fully equalize incomes


• Note that a 100 percent tax rate would lower average income

• Assume that each individual’s income has two components, one responsive to tax rates [yi (t)] and one not responsive (ni)

– also assume that the average of ni is zero, but its distribution is skewed right so nm < 0


• Now, utility is given byUi = (1-t)[yi (t) + ni] + b

• The individual’s first-order condition for a maximum in his choice of t and g is now

dUi /dt = -ni + t(dyA/dt) = 0

ti = ni /(dyA/dt)

• Under majority rule, the equilibrium condition will be

t* = nm /(dyA/dt)

The Groves Mechanism

• Suppose there are n individuals in a group– each has a private and unobservable

valuation (ui) for a proposed taxation-expenditure project


• The government states that, if undertaken, the project will provide each person with a transfer given by

i

ii vt

• If the project is not undertaken, no transfers are made


• The problem for voter i is to choose his or her reported net valuation so as to maximize utility

i

iiii vutuutility


• Each person will wish the project to be undertaken if it raises utility

• This means

0 i

ii vu

• A utility-maximizing strategy is to set vi = ui

The Clarke Mechanism

• This mechanism also envisions asking individuals about their net valuation of a public project– focuses on “pivotal voters”

• those whose valuations can change the overall evaluation from positive to negative of vice versa

The Clarke Mechanism

• For these pivotal voters, the Clarke mechanism incorporates a Pigouvian-like tax (or transfer) to encourage truth telling– for all other voters, there are no special

transfers

Important Points to Note:• Public goods provide benefits to

individuals on a nonexclusive basis - no one can be prevented from consuming such goods– such goods are usually nonrival in that

the marginal cost of serving another user is zero

Important Points to Note:

• Private markets will tend to underallocate resources to public goods because no single buyer can appropriate all of the benefits that such goods provide


• A Lindahl optimal tax-sharing scheme can result in an efficient allocation of resources to the production of public goods– computing these tax shares requires

substantial information that individuals have incentives to hide


• Majority rule voting may not lead to an efficient allocation of resources to public goods– the median voter theorem provides a

useful way of modeling the outcomes from majority rule in certain situations


• Several truth-revealing voting mechanisms have been developed– whether these are robust to the special

assumptions made or capable of practical application remain unresolved questions

PUBLIC GOODS Snyder and Nicholson, Copyright ©2008 by Thomson South-Western. All rights reserved.

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Transcript of PUBLIC GOODS Snyder and Nicholson, Copyright ©2008 by Thomson South-Western. All rights reserved.