©2005, Southwestern Slides by Pamela L. Hall Western Washington University Public Goods Chapter 22.

©2005, Southwestern

Slides by Pamela L. Hall

Western Washington University

Public Goods

Chapter 22

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Introduction Previous chapters generally considered only private

goods Commodities consumed individually by consumers

• One consumer’s consumption of such a commodity precludes other consumers’ consumption

This chapter considers nonrival commodities One consumer’s consumption of a commodity does not

preclude other consumers’ consumption• Called public goods

Goods where there is no congestion For example, one consumer’s satisfaction from breathing clean air

does not rival (compete with) other consumers gaining pleasure from also breathing the air

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Introduction We define public and private goods in terms of their rivalry and

exclusive characteristics We investigate exclusive but nonrival commodities and

condition for renting instead of selling nonrival commodities We address free-rider problem associated with public goods in a

game-theory framework Develop Pareto-efficient conditions for allocating public goods

We discuss how to obtain Pareto-efficient allocation when markets can be developed to establish prices for public goods (called Lindahl prices)

We discuss Clarke tax Provides a second-best Pareto-efficient mechanism for allocating public

goods

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Introduction

Aim in chapter is to understand distinguishing characteristics between private and public goods Why a free market (decentralized control) will

not result in a Pareto-efficient allocation of public goods• Leaves applied economists with task of

developing various mechanisms for determining optimal allocation of resources to production of public goods

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Rivalry and Exclusive-Good Characteristics

Public good: nonrival commodity One consumer’s consumption does not reduce amount available to other consumers Exists when marginal cost of another consumer’s consuming commodity is zero

Private good: rival commodity Depletable or diminishable commodity Each additional unit consumed by one consumer results in less of commodity

available for other consumers For a rival commodity, congestion is so severe only one consumer can consume

commodity Both public and private goods are further classified based on their exclusiveness

Exclusive commodity • Other consumers can be excluded from consuming the commodity

Nonexclusive commodity • Either it is illegal to exclude other consumers from consuming the commodity or cost of

exclusion is prohibitive

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Pure private goods are distinguished from private goods by their exclusiveness Private goods are all rival commodities Pure private goods have additional criterion of being

exclusive Similarly, public goods are all nonrival commodities

Pure public goods are also nonexclusive If consumption of a nonexclusive commodity also

does not deplete commodity for other consumers (a nondepletable commodity) Commodity is a pure public good

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Pure public goods are a specific type of externality that affects all consumers in an economy If one consumer is to consume a certain amount of a pure

public good• Then all consumers will consume that same level

Examples of commodities possessing characteristics of rivalry and exclusiveness are provided in Table 22.1

An exclusive commodity, such as food, with high congestion costs associated with rivalry is a pure private good Cost of an additional unit of commodity is nonzero

• In a free-market economy, such pure private commodities are generally provided by firms

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Table 22.1 Rivalry and Exclusiveness in Commodities

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Table 22.1 also shows that a private good, such as fire protection, can also be a nonexclusive commodity In contrast, public goods with zero congestion costs are nonrival

• An additional consumer does not add any additional cost for providing commodities

For exclusive public goods, consumers can be excluded unless, for example, they pay an entrance price (subscription fee, toll, or ticket) And marginal cost of an additional subscriber is zero

• Exclusion can also be based on some nonprice criteria Such as gender, race, national origin, or social status

However, in U.S. such criteria are generally illegal

In free markets, public goods can be privately produced by firms (such as movie theaters) Or publicly produced by government agencies (such as toll roads and bridges)

• Nonexclusive pure public goods (such as street lights) are generally provided solely by government agencies

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Distinctions among rival, nonrival, exclusive, and nonexclusive commodities are in terms of degree to which a commodity falls in one category or another For example, fire and police protection could be considered pure public goods

for a community • If protection is at a level where marginal cost of an additional consumer is near zero

As number of consumers increases with an associated increase in marginal cost Fire and police protection would then become more rival commodities

Increased highway congestion at some point will raise marginal cost of highway commuting

• Increases degree of rivalry on roads (Labor Day Weekend)

Finally, consumer preferences for a public commodity are not always positive Bad public commodities (called public bads) also exist

• Examples are environmental degradation (including air and water pollution), global warming, and species extinction

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Nonrival but Exclusive Commodities (Public Goods)

Nonrival characteristics of certain commodities allow Public libraries to share books and electronic

media Video outlets to share (for some rental fee)

videos and electronic games Equipment rental firms to rent a variety of items

from backhoes to party supplies

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Consider case of a firm offering commodity assuming no sharing exists

Let p(q) be inverse demand function with associated constant marginal and average cost of c Firm’s profit-maximizing problem is

If consumers rent commodity instead of purchasing it Level of consumption, y, will be greater than level of

production• For example, more videos will be watched than produced

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Let be number of times each commodity is shared by a consumer Level of consumption is y = q

Assuming all commodities are only rented, rental price would be p(y) Along with some positive transactions cost, ct, of renting versus

owning commodity• Purchasing a commodity reduces transaction costs involved with renting

Major reason why many households purchase rather than rent lawn mowers

Marginal consumers, with zero consumer surplus, would be willing to pay p(y) to rent commodity minus this transactions cost, p(y) - ct

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Further assuming that there are of these marginal consumers sharing one commodity Per-unit price firm receives is times p(y) - ct

[p(y) – ct] = [p(q) – ct]

Firm’s profit-maximizing problem for renting commodity is

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Comparing this profit-maximizing problem for producing and then renting output versus maximizing problem associated with producing and then selling output Only difference is in marginal costs

• Marginal cost for selling commodity is c

• Marginal cost associated with producing commodity for renting is [(c/) + ct]

Profits will be higher for renting if marginal cost for production associated with renting is lower than marginal cost for selling

c/ + ct < c

Illustrated in Figure 22.1

Assume demand for renting is same as for purchasing Both can be represented by same demand curve

• For profit maximization, firm will equate MR to MC

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Figure 22.1 Renting versus selling a product

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Marginal cost for renting is lower than marginal cost for selling Firm will set a rental price of pR*and y* of

commodity will be consumed• If firm sold product, price and quantity sold would be

p* and q*

• If marginal cost for renting is less than marginal cost for selling

Profits from renting π[(c/) + ct], are higher than for selling, π(c)

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The more a product is characterized as nonrival The higher will be the number of times each product is shared with

consumers, As increases, marginal cost of renting falls relative to selling

• Enhances profitability of renting versus selling product

As approaches infinity, condition for renting instead of selling commodity is ct < c Firm will then rent commodity instead of selling it

• When marginal cost of production is greater than consumers’ transactions costs of renting instead of owning

For example, transactions cost for newly released videos is relatively low compared with marginal cost

Consumers generally will rent new releases In contrast, marginal selling cost of used videos (salvage value) is relatively low

Consumers may instead purchase video rather than rent it

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Free-Rider Problem Nondepletable and nonexclusive characteristics of a

pure public good (such as PBS) Result in each consumer’s purchase of commodity

providing utility • Not only directly to this consumer but also to all other consumers

If consumers only consider their own utility in purchasing commodity and not effects such purchases have on all other consumers Externalities are present

• Consumers have an incentive to let other consumers purchase public good and receive utility from it without any cost

Called free-rider problem

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Free-Rider Problem A free rider is a consumer who cannot be excluded from receiving benefits

of a nondepletable commodity But is unwilling to pay his portion of cost (a non-PBS member viewing a PBS

program) By not cooperating and paying his portion of cost associated with public

good, free rider gains Another form of Prisoners’ Dilemma game

As an illustration, assume two roommates with public good of a clean kitchen (Table 22.2) If they cooperate and both agree to share in cleaning, their payoffs are 50 each

• Payoffs could be in monetary units or some other measurement

However, by not cooperating, becoming a free rider, and letting the other roommate clean

• Free rider can increase her payoff from 50 to 120 Nash-equilibrium result is both attempting to be free riders (not cooperating)

Results in a dirty kitchen

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Table 22.2 Free-Rider Problem for a Clean Kitchen as the Public Good

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Free-Rider Problem

Generally, in case of a small number of agents with limited or no transaction costs Coase Theorem applies and this externality

problem is resolved However as number of agents increases, a

Coasian solution is generally not possible With a large number of agents, it is generally

easy to be a free rider

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Pareto-Efficient Conditions for Pure Public Goods

Efficient allocation of a pure public good Sum of each agent’s willingness-to-pay is equal to cost of

public good

Recall that a Pareto-efficient allocation condition for consumer 1 considering purchasing commodities x1 and y is

MRPT denotes marginal rate of product transformation MRS is marginal rate of substitution

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Letting x1 and y be a private good and a pure public good, respectively MRS1(x1 for y) is how much consumer 1 is willing to sacrifice of the private

good, x1, for one more unit of pure public good, y

• MRS1(x1 for y) is consumer 1’s maximum willingness-to-pay, or reservation price, for pure public good

However, condition does not consider externalities associated with pure public good With these externalities, society’s maximum willingness-to-pay (MRSS) is

higher• Given that pure public good y provides same positive benefits to other consumers

MRPT(x1 for y) = MRS1(x1 for y) < MRSS(x1 for y)

Level of pure public good provided in a perfectly competitive market is below socially efficient solution

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Can develop Pareto-efficient condition for a pure public good supplanting inefficient condition, MRPT = MRS1

By considering a two-consumer economy with purchasing decisions of x1 and y

Let y be amount of pure public good and x1 and x2 be amounts of private good associated with consumers 1 and 2, respectively

• There is no subscript on y

Both consumers consume same amount of y, and y is nondepletable• However, they may consume different amounts of private commodity

So x1 + x2 = Q

Where Q is total amount of commodity produced

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Will describe technological possibilities of this economy by production possibilities frontier, f(Q, y) = 0 Welfare-maximization problem is

• Where U1 and U2 are utility functions for consumers 1 and 2, respectively

• Ц is some social-welfare function

Forming the Lagrangian

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F.O.C.s are

Q/x1 = Q/x2 = 1, given Q = x1 + x2

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Solving for Lagrangian multiplier and equating yields

The last equality establishes

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Marginal gain in welfare associated with additional consumption of commodity Q by a consumer must be equal for all consumers If this is not the case, it would be possible to reallocate Q

among consumers in a way that increases social welfare

Cross-multiplying first equality yields

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First term is consumer 1’s MRS1(x1 for y)

Second is consumer 2’s MRS2(x2 for y) Term on right-hand side is MRPT(Q for y) between

public and private good Thus, condition for Pareto efficiency is

MRS1 + MRS2 = MRPT

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Instead of perfectly competitive condition, MRS1 = MRS2 = … = MRSn = MRPT for n consumers Pareto-efficient condition is

Sum of willingness-to-pay (MRSS) equated to cost (MRPT) results in a Pareto-efficient allocation of pure public good

An example is if a home theater system costs $5000 and 100 sorority sisters are each willing to pay $50 Individually, no one sister would purchase the system

• But collectively Pareto-efficient response would be to purchase it MRSS is sum of individual consumers’ MRS

Accounts for benefits all consumers receive from pure public good Equate MRSS to MRPT to determine Pareto-efficient level of resource allocation

Individual consumers’ MRS(xj for y) are each consumer’s reservation price Maximum willingness-to-pay for pure public good

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Can relate concept of MRS(xj for y) as a consumer’s reservation price for y to market price for y, py

By letting price of Q be a numeraire, so pQ = 1

For utility maximization consumer sets MRS(xj for y) = py/pQ

For pQ = 1, a consumer’s reservation price is equal to market price, MRS(xj for y) = py

• Thus, for a pure public good, summing reservation prices yields total per-unit price society is willing to pay for pure public good y

Equating total per-unit price to cost of supplying one more unit, MRPT(Q for y), yields a Pareto-efficient allocation

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Consumers paying their reservation price per unit for pure public good is one Pareto-efficient outcome Yields a marked distinction for efficiency between private

and pure public goods

For a private good, all consumers consume different amounts of commodity but pay same market price

For a pure public good all consumers consume same amount of commodity (say, national defense) but pay different prices Illustrated in Figures 22.2 and 22.3

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Figure 22.2 Horizontal summation of the demand curves for a private good

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Market demand curve for a private good is horizontal summation of individual consumers’ demand curves for private good

Treating reservation price, MRS(y for xj), as price of private good Q, Pareto-efficient allocation is for both consumers to pay the same price MRS1(y for x1) = MRS2(y for x2) = MRPT(y for Q)

From Figure 22.2, at this price, 10 and 8 units of Q are demanded by consumers 1 and 2, respectively Yields a total market demand of 18

• Individual demand curves are based on preference orderings of consumers

Represented by indifference curves

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For a public commodity, each consumer consumes same amount of commodity but at a different price

Derive market demand curve by vertically summing individual consumers’ demand curves Shown in Figure 22.3

• Consumer 1’s MRS1(x1 for y) is 4 and consumer 2’s MRS2(x2 for y) is 1

Pareto-efficient allocation is where MRS1(x1 for y) + MRS2(x2 for y) = MRPT(Q for y)

If price of Q is numeraire, pQ = 1, then ratio py/pQ = 4 = MRS1(x1 for y) and py/pQ = 1 = MRS2(x2 for y) Consumer 1 pays $4 per unit for pure public good and consumer 2 pays

$1• But they each consume the same amount

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Figure 22.3 Vertical summation of the demand curves for a pure public good

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One solution for inefficiency of perfectly competitive markets in providing for pure public goods Establish another market that will account for externalities associated with

public goods Offered such a solution in Chapter 21

Market for permits could be established to yield a second-best Pareto-efficient allocation

• Solution may or may not be feasible, depending on nature of inefficiency A market solution works well when a commodity can be segmented

Proportion of property rights for commodity can be transferred from public to a private agent

Market-permit system is one case where this market solution can be feasible Permits transfer a proportion of a common property commodity to a private

agent• However, it is not as attractive for correcting public-goods allocation problem

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Nonrival and nonexclusive characteristics of a pure public good prevent segmenting of commodity For example, an individual household cannot purchase a

proportion of national defense To establish such a market (a Lindahl market) for a pure

public good Each consumer would have to voluntarily reveal and pay their

reservation price (their Lindahl price) per unit for a pure public good

• Summing reservation prices and equating sum to MRPT would determine efficient allocation of pure public good

• Such markets generally are not feasible Mainly as a consequence of free-rider problem

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Consumers’ dominant strategy Understate their preferences (by discounting their Lindahl prices) and rely on other

consumers to pay a larger share for pure public goods Nonexclusive characteristic of pure public goods fosters this free-rider strategy One solution to free-rider problem

For government to impose a per-unit tax on each consumer equivalent to their respective Lindahl prices

• However, unlike reservation prices for private goods, Lindahl prices are not revealed in market

Consumers cannot adjust their level of consumption unilaterally Destroys possibility of a market for pure public goods

• Government agency has no feasible mechanism for determining each consumer’s willingness-to-pay

To impose such a tax government agency must perfectly price discriminate among consumers Such systems are difficult to achieve

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Even if it were feasible to determine consumers’ Lindahl prices and perfectly price discriminate Consumers may object to paying differentially per unit for pure public

goods• May be more inclined to support funding for pure public goods based on

ability to pay rather than willingness-to-pay

Many public health and housing agencies base fees and rents on ability to pay

In general, pure public goods are financed by taxes based on income and wealth As opposed to decentralized control for allocation of private goods

• Some type of centralized control is required for public-goods allocations Determination of types, amounts, and funding for pure public goods may then

be based on some mechanism design

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In general, such mechanism designs attempt to determine intensities of individual and group desires And formulate a mechanism composed of

policies and rules for group choice and actions Clarke tax is one such mechanism

• Under some rather restrictive conditions, tax provides incentives for consumers to reveal their true preferences for a social choice

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Clarke Tax

Eliciting truthful preferences for pure public goods can mitigate misallocation of governments’ taxing, spending, and regulatory authorities By overcoming free-rider problem

Proponents of Clarke tax mechanism claim, based on a second-bid auction, tax mechanism will not completely cure free-rider problem by yielding a Pareto-efficient allocation But has potential to treat the symptoms

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Clarke Tax As an illustration, consider a group of consumers jointly

deciding on purchase of a pure public good If purchased, every consumer will pay a predetermined amount for

purchasing the good• An example is an appliance such as a microwave oven in a dormitory

Let cj be this predetermined amount for consumer j

• Summing over all consumers equals cost of pure public good Consumers will then state how much they are willing to pay

Difference between consumer j’s willingness-to-pay, WTP j, and predetermined amount is net benefit, NB j

NBj = WTPj - cj

If sum of net benefits over all consumers is positive Then pure public good should be purchased

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Clarke Tax Problem is designing a mechanism that provides an incentive for

consumers to reveal their true net benefit Instead of revealing an exaggerated figure in an attempt to influence this

social choice However, an exaggeration is only of concern if it affects social choice For example, say consumer j attempted to be a free rider by stating a

value of zero Yielding a net benefit of -cj

• If sum of net benefits over all consumers is still positive Free rider does not influence social choice, so it is of no concern

Only consumers whose exaggeration will affect social choice are of concern Such consumers are called pivotal consumers

• Their net benefit determines whether sum of net benefits is positive or negative In the extreme, all consumers could be pivotal consumers or none could be pivotal

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Clarke Tax It is possible that any one consumer could be pivotal

Ensuring that all potential pivotal consumers have the right incentives corresponds to ensuring that all consumers reveal their true preferences

When a social choice is changed by a pivotal consumer Adversely affects other consumers

For example, if other consumers have positive net benefits for a pure public good and pivotal consumer’s negative net benefit resulted in not purchasing good All other consumers are made worse off

A measure for how much other consumers, in aggregate, are worse off Sum of net benefits excluding pivotal consumer, say, consumer 1

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Clarke Tax If other consumers have negative net benefits for pure

public good and pivotal consumer’s positive net benefit resulted in purchasing commodity All other consumers are again made worse off

• Measure for how much other consumers, in aggregate, are worse off is the negative of the sum of net benefits excluding pivotal consumer, say, consumer 1

Analogous to imposing a Pigouvian tax on negative externalities, pivotal consumer is taxed by amount he or she harms other consumers, 1

• Called a Clarke tax Which is paid by all pivotal consumers

Results in these consumers having incentive to reveal their true preferences for pure public good

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Clarke Tax Clarke tax mechanism is a second-bid, sealed bid auction for a pure

public good A pivotal consumer’s tax is equal to second-highest valuation

• Sum of all other consumers’ net benefits Benefits from tax revenue cannot be distributed to other consumers in such a manner

that it influences other consumers’ net benefit for pure public good

Consumers facing a higher tax rate relative to others may not respond well to tax-discriminating nature of Clarke tax

One problem with Clarke tax is that it is not necessarily Pareto efficient Predetermined payment may result in some cases where a group of

consumers has negative net benefits for pure public good• Even when sum of net benefits is positive

Purchasing pure public good will harm these consumers Not result in a Pareto improvement Some type of compensation principle would be required to justify social-welfare

benefits of a Clarke tax

©2005, Southwestern Slides by Pamela L. Hall Western Washington University Public Goods Chapter 22.

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