Download Pareto-Efficient Conditions for Pure Public Goods

Public Goods
Chapter 22
Slides by Pamela L. Hall
Western Washington University
©2005, Southwestern
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
2
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
3
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
4
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
5
Rivalry and Exclusive-Good
Characteristics

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
6
Rivalry and Exclusive-Good
Characteristics

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
7
Table 22.1 Rivalry and
Exclusiveness in Commodities
8
Rivalry and Exclusive-Good
Characteristics

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
9
Rivalry and Exclusive-Good
Characteristics

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
10
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
11
Nonrival but Exclusive
Commodities (Public Goods)



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
12
Nonrival but Exclusive
Commodities (Public Goods)


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
13
Nonrival but Exclusive
Commodities (Public Goods)

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
14
Nonrival but Exclusive
Commodities (Public Goods)

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
15
Figure 22.1 Renting versus
selling a product
16
Nonrival but Exclusive
Commodities (Public Goods)

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)
17
Nonrival but Exclusive
Commodities (Public Goods)

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
18
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
19
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
20
Table 22.2 Free-Rider Problem for a
Clean Kitchen as the Public Good
21
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
22
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
23
Pareto-Efficient Conditions for
Pure Public Goods

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
24
Pareto-Efficient Conditions for
Pure Public Goods

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
25
Pareto-Efficient Conditions for
Pure Public Goods

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
26
Pareto-Efficient Conditions for
Pure Public Goods

F.O.C.s are
 Q/x1 = Q/x2 = 1, given Q = x1 + x2
27
Pareto-Efficient Conditions for
Pure Public Goods

Solving for Lagrangian multiplier and equating
yields

The last equality establishes
28
Pareto-Efficient Conditions for
Pure Public Goods

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
29
Pareto-Efficient Conditions for
Pure Public Goods




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
30
Pareto-Efficient Conditions for
Pure Public Goods

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
31
Pareto-Efficient Conditions for
Pure Public Goods


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
32
Pareto-Efficient Conditions for
Pure Public Goods

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
33
Figure 22.2 Horizontal summation of the
demand curves for a private good
34
Pareto-Efficient Conditions for
Pure Public Goods



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
35
Pareto-Efficient Conditions for
Pure Public Goods


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
36
Figure 22.3 Vertical summation of the
demand curves for a pure public good
37
Pareto-Efficient Conditions for
Pure Public Goods

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 Paretoefficient 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
38
Pareto-Efficient Conditions for
Pure Public Goods

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
39
Pareto-Efficient Conditions for
Pure Public Goods

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
40
Pareto-Efficient Conditions for
Pure Public Goods

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
41
Pareto-Efficient Conditions for
Pure Public Goods

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
42
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
43
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, WTPj,
and predetermined amount is net benefit, NBj
 NBj = WTPj - cj
If sum of net benefits over all consumers is positive
 Then pure public good should be purchased
44
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
45
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
46
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
47
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
48