Public Goods
Topics in Political Economy
Ana Fernandes
University of Bern
Spring 2010
1
Public Goods
• Earlier in the course, we saw that markets achieve efficiency for private
goods — goods that benefit one person
— When goods are nonrival in consumption and can be enjoyed by many
simultaneously, we do not expect an efficient outcome
— Because many people benefit simultaneously, public goods could well
be called collective goods
— The collective nature of benefits suggests the need for collective means
of decision making, such as voting
• We will now briefly look into public goods and their provision
— Next, we will address voting as a means of reaching collective decisions
with regard to the provision of public goods and redistributive taxation
1.1
Properties of Public Goods
Benefits from Private and Public Goods
• For a private good, an amount consumed by one person is not available
for the consumption of others
— If individual i consumes amount xi ≥ 0 and X is the total quantity
available of a private good, it follows that:
X=
n
X
xi
i=1
— Private goods are therefore divided or distributed among the population
— Consumption of private goods is rival in that amount xi consumed by
individual i cannot be consumed by anyone else (e.g. a hamburger)
• For public goods, no such division takes place
— Total quantity G of a public good is also the quantity gi available to
each of the n individuals:
G = g1 = g2 = ... = gn
The quantity of a public good available to one person is also the quantity available to everybody
— This means that public goods are nonrival in consumption: consumption by one individual does not diminish or lessen the consumption of
others (e.g. national defense)
• Although the amount consumed is the same, how individuals value that
amount of course varies
— For example, when a Mozart symphony is played, some people will
enjoy it, some will prefer Jazz and others still would prefer silence...
Characteristics of Pure and Congestible Public Goods
• Public goods may be subject to congestion
— For example, highways are a public good up to the point where additional cars affect the usage of other vehicles already on the highway
— If traffic is too heavy, there will be traffic jams or cars will have to slow
down
• We say that public goods are pure when
Individual benefits from a public good are independent of the number of
users
• When individual benefit declines with the number of users (or after the
number of users goes above some threshold), the public good is said to be
congestible
• Figure 3.1a shows how individual benefit and the number of users are
related for a pure public good
— In 3.1b we see the same relationship for a congestible public good
— There, the public good is pure up to a number n1 of users, after which
congestion begins and individual consumption declines
— When more than n2 consumers use the public good, individual benefit
is negative
— When more than n2 cars are on the highway, people would be prepared
to pay to have other drivers leave the highway
• When public goods are congestible, it is usually possible to exclude people
from consumption
— In the highway case, for example, toll booths can be set to charge
people from using the highway
— The toll can be set high enough so as to reduce the number of users
to a level considered adequate
• The possibility of exclusion allows markets to be created in which people
pay in order to benefit
— The government therefore has no necessary responsibility in the supply
of public goods
— As in the case of gyms, movie theaters, concerts, sports events, transportation, private schools, private hospitals and so on, in general,
Congestible public goods can be privately and competitively supplied
• Since markets for congestible public goods resemble markets for private
goods, we will focus our attention on pure public goods (or on congestible
ones when the number of users is below the congestion threshold)
Exclusion
• People cannot always exclude themselves from the consumption of pure
public goods nor it is always possible to exclude people (for example in the
cases of the benefits from national defense or the rule of law)
— However,
Where exclusion is possible, exclusion from a pure public good is inefficient
• We see this by examining the total benefit to n people from a pure public
goods:
Wn =
n
X
i=1
Bi − C
• When an additional person benefits, the fixed cost C of supplying the
public good remains unchanged
— However, with n + 1 beneficiaries, total benefit from the public good
increases to:
Wn+1 =
n+1
X
i=1
Bi − C
— Thus, the change in total benefit from the public good is the personal
benefit to the additional person:
∆W = Bn+1 > 0
— Therefore,
Allowing an additional person to benefit from an available public good
is Pareto-improving: someone is better off and no one is worse off
• The marginal cost of use of an existing public good by an additional person
is zero:
M C use = 0
— Efficient use requires that
MB use = M C use
— Therefore, efficient use requires free use or free access to existing public
goods:
P use = 0
• The cost of supplying a public good differs from the cost of use
— A bridge is costly to build
— However, once the building costs have been incurred and it has been
built, the cost of use of a noncongested bridge is zero
— The efficient price of use of the bridge is therefore zero
Efficient use requires that access to pure public goods be free
The Dilemma of Exclusion
• Though efficiency requires that access to public goods be free, their provision must still be financed
— If people are able to benefit from pure public goods once they are
provided, they have no incentive to contribute to the costs of their
provision
— If feasible, exclusion might then be exercised as a means of giving
people incentives to contribute to public good provision
— For example, if fire fighters only put off fires of houses of people who
contribute to the fire squad, people have an incentive to do so
— Although exclusion is inefficient as seen earlier, it might be the means
for giving people incentives to pay for public goods
1.2
Voluntary Personal Payment for Public Goods
• We next address the question:
How effective can a society be in supplying public goods without a government that has the authority to tax?
• Figure 3.3a shows demand for a private good
— Total market demand is the horizontal sum individual demands M B1
and M B2 of two people
— At the market equilibrium price P E , person 1 purchases quantity q1∗
and person 2 the quantity q2∗
³
— At price P E , market demand is Q∗ = q1∗ + q2∗
´
• Figure 3.3b shows the total demand for a public good, defined in terms of
either quantity or quality
— For a public good, the quantity or standard available to one person is
also that available to everyone else
— Therefore, collective (or total) demand for a public good is found by
asking:
How much are all beneficiaries of a public good willing to pay in total
for a given common quantity or standard?
• In figure 3.3b, person 1’s marginal willingness to pay for the public good
at common quantity G∗ is P1∗
— Person 2’s marginal willingness to pay is P2∗
³
´
— Combined marginal willingness to pay at quantity G∗ is P1∗ + P2∗ =
P∗
• Point F in figure 3.3b is on the combined demand function
persons 1 and 2
P
MB of
— The combined or collective demand function is found by vertically summing M B1 and M B2 at different quantities or standards of the public
good
• For private goods, as the amount of a good consumed by one individual is
not available for the consumption of another, the summation is horizontal
— At a given price, we add up the quantities demanded by different individuals and which will be consumed by each individual on his own
• For public goods, the quantity consumed is the same across people
— Therefore, for a given amount G of public good provided, we add up
the valuation that different individuals have of that quantity
— It makes no sense to add horizontally individual marginal benefit curves
since the same common quantity G will be available to all
Supply
• There is no difference between supply functions for private and public goods
— As for private goods, the inputs required for the production of public
goods are generally competitively supplied in markets
Efficient Voluntary Payments for Public Goods
• Figure 3.4 combines a competitive supply function for a public good with
P
total willingness to pay, MB
— At point E, where quantities demanded and supplied are equal, payment of price P E of the public good is shared
— Person 1 pays the personal per-unit price P1E and person 2 pays P2E
• Personal per-unit prices in figure 3.4 are determined by each person’s marginal benefit at the equilibrium quantity GE
— We see that
P1E = M B1,
P2E = MB2
• Adding up the personal per-unit prices and personal marginal benefits from
the previous expression, we get:
P E = P1E + P2E =
X
MB
(1)
i
— Because the inputs for public goods are competitively supplied:
P E = MC
(2)
— Combining (1) and (2), we get that, at point E,
X
MB = MC
(3)
i
— Condition (3) is satisfied when people voluntarily pay for public goods
according to personal benefit
• Condition (3) also ensures efficient provision of public goods as follows
— Total benefit from the public good is
W =
X
i
B−C
— Total benefit is maximized when
X
M Bi = MC
i
— Condition (3) describing the outcome of voluntary payments at point
E is therefore the condition for maximum W
• We conclude that:
Voluntary payment for public goods according to personal benefit maximizes W and so is efficient
• There was no reason to expect otherwise
— Personal payment in competitive markets is efficient for private goods
— The only difference regarding public goods is that total demand is
derived by vertical — rather than horizontal — summation of individual
demands
— Of course we are assuming that people have an incentive to pay for
public good provision
Asymmetric Information and Under-Supply of Public Goods
• For private goods, individuals face a common market price and choose
different personal quantities
— For public goods, individuals have the same quantity but are asked to
pay different personal prices based on personal benefit
— Personal benefit, however, is subjective private information
— Personal payment for public goods thus involves asymmetric information: only people themselves know their private benefit
• By taking advantage of asymmetric information and understating true personal benefit, people can reduce the personal price they pay
— Figure 3.5a shows the personal gain from understating true marginal
benefit
— By claiming lower than true M B, the person in figure 3.5a reduces the
personal price when the quantity supplied is G∗ from P1 to P2
• In a competitive market for private goods, there can be no personal benefit
from misrepresenting personal preferences or willingness to pay:
— By misrepresenting preferences for private goods, people can only reduce their benefit (by getting an amount which is less than what they
would like to consume at the given price)
— But misrepresentation of M B for public goods when payment is voluntary leads to a reduction in the personal price paid
• The consequences of deception by only one person in a large population is
negligible
P
— Figure 3.5b shows the downward shift in the combined (untrue M B)
function of the population when many people deceptively claim low
personal benefit
— The quantity G0 of the public good that is provided is less than the
efficient quantity GE
— We conclude that
Personal incentives to understate benefit suggest that voluntarily financed public goods will be under-supplied relative to the efficient
quantity or standard
• The term free riding describes relying on others to pay
Under-supply of voluntarily financed public goods arises because of free
riding incentives
The Ideal Lindahl Consensus
• The ideal efficient outcome for voluntarily financed public goods is known
as the Lindahl solution, achieved through a Lindahl mechanism
— In the Lindahl solution, people are described as paying a share of the
cost of public goods according to their true preferences
— No one free rides
— The Lindahl solution emphasizes the need for consensus on the quantity
or standard of the public good
Ideally, people should not be compelled to pay for a quantity or standard
of a public good that they do not want; therefore, it is desirable that
there be consensus about the quantity or standard
• We previously described a market for public goods by expressing total
demand as the sum of MBs and supply in terms of the MC of inputs
— To demonstrate the Lindahl solution, we now change how we look at
demand and supply
— In figure 1, the horizontal axis measures the common quantity of a
public good
— No one is excluded from benefit so there is an efficient price of use of
zero
— Still, there is the need to pay for the public good
• The public good is competitively supplied at given market price P
— Person 1 pays a share s1 of the price P of the public good
— Therefore, person 1’s cost share or personal price is determined as
P1 = s1P
• The schedule OO in figure 1 is person 1’s demand expressed as a function
of his share s1
— The higher the share s1 paid by person 1, the lowest the quantity of
the public good desired by this person
— The schedule LL is the demand of person 2, also expressed as a function player 1’s share
— Because shares add to unity in this two person environment,
s2 = 1 − s1
— Therefore, the greater the share of the price paid for by person 1, the
more quantity of the public good is desired by person 2
• At s1 = 1, person 1 is paying fully for the provision of the public good
— He then wishes to consume a small amount of it
— Person 2, on the other hand, wishes to consume a lot, and, in particular,
a greater amount than person 1
• When s1 = 0, it is person 2’s turn to pay for the public good
— Person 1 wishes to consume more of the public good than person 2
does in this case
— From the continuity and monotonicity of demand functions, they must
cross only once
• The point E where they cross determines both the equilibrium quantity of
public good provision and the way the cost is shared
• Figure 1 can be interpreted in terms of demand and supply:
— Demand of one person is supply for the other
Because a public good provides collective benefit no matter who pays,
payment by one person for the public good is free supply for the other
• In the Lindahl solution, shown in figure 1, there is consensus about demand
for the public good
— That is, given share s∗1 that person 1 pays of P and the corresponding
s∗2, both persons agree on the quantity of the public good that ought
to be provided, G∗
— The Lindahl solution is therefore characterized by the consensus quantity demanded G∗ and cost shares s∗1 and s∗2
• We next show that the Lindahl-consensus solution results in the supply of
an efficient quantity or standard of the public good
— We confirm that supply of the consensus quantity G∗ is efficient by
showing that the Lindahl solution replicates the efficiency condition for
public good supply:
X
(4)
M Bi = MC
i
— In figure 1, at quantity or standard G∗,
Pi = s∗i P = MBi (G∗) ,
i = 1, 2
— Summing for the two people, we have:
(P1 + P2) = (s∗1 + s∗2) P =
X
MB
— With P =
of competitive supply of the public good and
³ MC because
´
∗
∗
because s1 + s2 = 1, the efficiency condition (4) follows from the
previous equation
— Therefore:
The Lindahl voluntary-payment consensus results in efficient supply of
public goods through voluntary personal payments
• However, the efficient Lindahl solution requires people to be truthful about
their benefits expressed in the willingness to pay curves
— In figure 2, understatement of personal benefits of person 1 changes
the determination of personal cost shares from point E to E’
— At E’, person 1 has a lower cost share than at point E and, consequently, the cost-share of person 2 has increased
— Person 2 confronts the same incentive to misrepresent personal benefit
— The mutual incentives to understate true benefit once more result in
inefficient under-supply of the public good
Strategic Behavior
• Understatement of personal benefit introduces strategic consideration
— Person 1 would not want to understate benefit too much
— Excessive understatement of personal benefits may push the other person’s contribution beyond a level where he would no longer want to
continue making contributions
— Likewise, if person 2 were to raise person 1’s contributions beyond
an acceptable level for person 1, person 1 would stop contributing to
financing the public good
The Lindahl Solution as a Benchmark
• The Lindahl solution is a normative portrayal of ideal truthful behavior of
beneficiaries of public goods
— People are thus viewed as trusting and trustworthy
— The Lindahl solution is, however, contrary to the premise that people
maximize utility or do the best for themselves
— By misrepresenting their preferences and thus reducing their cost shares,
people would be better off if taking the behavior of others as given
• We shall not assume that the Lindahl solution is feasible
— We can nonetheless use the Lindahl solution as a benchmark for efficient payment and supply of public goods
— The question of interest becomes whether there is some way of achieving the efficient Lindahl consensus outcome
— In particular, we would like to know whether governments can replicate
the efficient Lindahl consensus outcome