note6

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©David M. Nowlan 1999
Introductory Note on Public Goods
According to the standard theory of fiscal federalism, higher levels of government – the federal or
the provincial levels – should have responsibility for redistributive policies, including
redistributive taxation, while the lower or municipal levels should focus on providing government
goods and services efficiently with minimal redistributive effects. From this perspective, the
ideal municipal tax would be a "benefits" tax, one that equated at the margin the tax cost of some
service to the value of the benefit received on a person-by-person basis.
Governments at all levels, including municipal governments, will normally provide goods and
services that are difficult to provide, at least to provide efficiently, through the private market.
These will typically be goods and services that, to some extent at least, residents share: roads,
clean air, pure water, police and fire services, parks, administrative services and so on. We can
begin to understand the difficulty of providing these goods and services in efficient amounts, and
the equally great difficulty of paying for them through a benefits tax, by considering the nature of
public goods and the conditions that are necessary theoretically for their efficient provision.
The simplest form of public good, sometimes called a "pure" public good, is a good or service
that, once it exists, is available to all without congestion. Its benefits may have a geographic
dimension, i.e., they may be exhausted beyond some range or distance, and different people may
have very different views about the value of the benefits to them, but having more people enjoy
the benefits of the public good does not diminish its value to the rest. Because of this feature,
public goods are said to be "nonrivalrous" in consumption. Private goods by contrast are
"rivalrous." With respect to a private good, what I have you can't have; that is why they are
called rivalrous. With a public good, what I have, you can have too; they are nonrivalrous.
Examples of pure public goods of this sort are not easy to find, since most shared goods are
subject to congestion, and if more people are to get the benefit of a congested good, then more of
the good will have to be provided if the benefits to the rest are not diminished. Perhaps, at the
municipal level, clean air or a good water supply come close to being pure public goods, but
within limits we might also think of uncongested roads or uncongested parks and perhaps good
administrative leadership as being close to what we could call pure public goods.
Rare or not, let us continue to look first at the allocation of resources to pure public goods. We
can back into this by recalling the conditions for efficient resource allocation to private goods.
For some given rivalrous private good -- a bottle of beer, a loaf of bread – the efficiency
conditions are that every consumer's marginal valuation, MV, of the good must be the same and
this common MV must equal the marginal resource cost of the good, the MC. If we had a set of
consumers, A, B, C and so on then the efficient allocation of resources to private good X requires
that MVA = MVB = MVC = …… = MCX .
If you think about it, you'll appreciate that a competitive market achieves exactly this result.
Every consumer and every producer is a price-taker; the consumers adjust their consumption
amounts, shown by their demand curves, so that the price just equals each persons MV of the
good, and the producers adjust their supply so that the price equals their marginal cost. Thus, the
price functions as a signal, signalling forward to the consumers who adjust demand and backward
to producers who adjust supply. In equilibrium, there is just one price and it serves to bring MVs
and MCs into line. Thinking in terms of demand curves, you will recall that the market demand
curve for a private product is just the summation horizontally of each consumer's demand curve,
and the equilibrium price will be where this market demand curve cuts the market supply curve.
This price then reflects back to each consumer's demand curve and determines quantity
consumed.
The efficient allocation of resources to pure public goods looks quite different. Because a pure
public good is available to benefit everybody, we must now compare the sum total of everybody's
marginal valuation of the good to the marginal resource cost of producing the good. [Notice that
even if the benefits of the good are exhausted beyond some geographic range – say outside the
municipal jurisdiction – we can still write about its being available to "everybody" and simply
assign a zero MV to anybody who is beyond the effective range of the good or service.] This is
equivalent to writing, for our hypothetical community of person A, person B, person C etc., the
following as the efficiency condition: MVA + MVB + MVC + …. = MCG, where "G" is our pure
public good.
This efficiency condition may be illustrated by the accompanying diagram 1, which shows the
demand curves of three individuals, DA, DB and DC. These represent the marginal valuation
curves and the efficiency condition given above implies that the three curves should be added
vertically, in contrast with the horizontal addition of private-good demand curves, to get a
community demand curve, SUMD, which should equal the public good's marginal cost at point of
optimal provision, G* on the diagram.
Diagram 1
40
$
35
30
DA
DB
DC
SUM D
MC
25
20
15
10
tC
5
tB
tA
G*
0
1
2
3
G
amount of public good
(The representation of the optimality condition through the vertical summation of stable demand
curves is not completely general. Because of income effects, these demand curves may move
about, depending on the payment scheme that is implemented. Only if the income elasticity of
demand for the public good is zero will the demand curves remain stable. The first-order
optimality condition
 MV
i
 MCG is completely general; the vertical summation of demand
i
curves is a useful heuristic device to illustrate the condition.)
In order for a government to provide optimal amounts of a public good, the marginal valuations
of all members of the community for different amounts of the public good must be known, so the
government will know when the optimality condition is being met. There is no such need for
centralized knowledge in the case of competitive private-good markets. The price in a privategood market adjust up or down depending on whether there is excess demand or supply, and each
buyer and seller adjust his or her buying and selling action to the market price. This is said to be
an efficient decentralized mechanism for allocating resources to private goods.
As Samuelson noted in his 1954 article ("The Pure Theory of Public Expenditures," Review of
Economics and Statistics,vol. XXXVI, November 1954, 387-389), the basic problem of public
good provision is that there is no decentralized pricing or signalling scheme that can be used to
lead governments to efficient levels of public-good output. If people in the community are
simply asked about their preferences, as reflected in marginal public good valuations at different
levels of provision they will have strong incentives to give misleading answers, to lie about their
true preferences.
The type of bias in the community's answers depends on what the people think will be the
payment scheme for the public good. Suppose people are told, or believe, that they will be
charged an amount that reflects their own personal valuation of the good. In terms of the
accompanying diagram, this would mean that if persons A, B and C reported DA, DB and DC as
their respective marginal valuation schedules, then the authorities would provide G* of the public
good and charge tA per unit of the good to person A, tB to person B and tC to person C. In this
way the cost of the public good would be covered and each person would pay a "price" that just
equalled his or her stated marginal valuation.
Suppose DA, DB and DC are the true marginal valuation schedules. All three of the people, and
Person C in particular, have strong incentives to say that his or her marginal valuation is really
lower than the true schedules. If person C, for example, gave a schedule that was only half as
high as the true DC, then that person would only have to pay a tax cost per unit, tC, that, for each
level of public-good provision, was only one-half the amount that would have to be paid if he or
she gave the true DC schedule. Each person has the same incentive to understate his or her
schedule of marginal valuations.
You might think that the best strategy under these circumstances would be to say (falsely) that
you place no value whatsoever on the public good. However, if there are literally only two or
three people in the community, this might not be best. Notice that if you are one of these people
and if you give a lower-than-truthful valuation, the authorities will provide less of the public good
MVi schedule will be lower), whereas you would always like more of the public
(because the

i
good provided you didn't have to pay for it. So, with a small number of people, you may be
aware that your understating your true preference may have a noticeable affect on the amount of
public good provided, and this will limit the extent of your strategic untruthfulness. (As an aside,
the outcome of two or three people interacting to provide a public good can often be modelled as
a Nash equilibrium, much like the duopoly equilibrium outcome that you have studied in
microtheory.) If the community is very large, with thousands or tens of thousands of people, you
might reasonably feel that whatever you say is not going to affect in any noticeable way the level
of public good provision. In this case, you best apparent strategy will be to say that you place no
value on the good; your marginal valuation curve is horizontal at $ = 0. Of course, everybody
else will likely feel the same way, so the authorities would be faced with adding up many zero
valuation curves: they would be led to provide none of the public good. It is a classic prisoner's
dilemma situation. No one has an incentive to do other than lie, but everybody's lying leads to a
community outcome that is worse for everybody than if they had shared the cost in proportions
that honestly represented their different preferences. You can begin to see why Samuelson was
skeptical about the existence of a decentralized signalling mechanism.
The problem exists also if people are told or believe that what they pay for the public good will
bear no relationship to the marginal valuation schedule they volunteer. In that case, why not
claim that you place an enormous valuation on the public good, as long as you continue to get
some benefit from having more of it? In the limit, everyone would choose the amount of the
public good beyond which it delivered no value to them (i.e., where their demand curve hits the
horizontal axis) and say that they placed an extremely high valuation on that amount of the good
– essentially claiming a vertical marginal valuation curve at that point. The authorities, if they
followed the adding up process to arrive at an optimal amount of the good, would have to provide
an amount equal to the amount the most good-loving person gave, and resources would be very
overcommitted to this public good.
Mechanisms that aim to have people directly reveal their preferences for this public good seem
doomed to failure. Instead, very commonly members of a community will decide on the
appropriate amount of the public good through a voting procedure, either through a direct vote or
indirectly through their elected representatives. Just before leaving this introductory section, I
will comment upon the possibility of achieving an optimal allocation of resources to a public
good through such a voting mechanism. There is a theorem that says that optimal allocations are
possible, under some circumstances.
Suppose that every member of the community will contribute equally to the cost of the public
good and that the only question is, how much to provide? With three people in the community,
and with constant costs (a horizontal MC curve), each member would pay MC/3 per unit of
public good provided. This per-person cost is shown in the second diagram, above. Suppose a
vote is now taken of the following sort: the three people are first asked whether they prefer one
unit of the good or two. The winning amount would then be tested against 3 units of the public
good.
In a vote of one unit versus two, clearly two units will win. Look at diagram 2. Person A prefers
one unit to two, but both B and C would prefer two units to one. (Remember that they are all
paying MC/3 dollars per unit, so you just have to see what the demand or marginal valuation
schedules say about each person's demand at that "price.") Now ask for their opinion on two
units versus three. In this case, two will again win: persons A and B favour two; only person C
favours three units.
Diagram 2
40
35
30
25
DA
DB
DC
SUM D
MC
MC/3
MC
$ 20
15
10
MC/3
5
G*
0
1
2
3
4
amount of public good
Notice that two units is exactly the preferred amount of person B (given the price of MC/3 per
unit). Person B is the "median" voter, the voter who, with respect to preferences, has an equal
number of people on either side; in this case, at any price, one person always prefers more than
person B and one person always prefers less. As a general matter, in any pairwise comparison of
possible amounts of public good, the preferences of such a median voter will always win the vote
and become the chosen option. This is the "median voter theorem."
An extension of this theorem goes on to say something quite remarkable: that if the voters
preferences are distributed symmetrically around the preferences of the median voter, than the
vote-winning amount of public good will also be the efficient or optimal amount. To see this,
look again at the diagram and look at the marginal valuations for two units of the public good.
This is the amount chosen by the voting procedure described above. Now, if person A's marginal
valuation lies the same amount below person B's as Person C's lies above, then two units is
exactly the amount of public good where the sum of the demand curves cuts the MC curve. In
general, such a symmetry of marginal valuations above and below the median valuations will
result in the median voter's choice also being the efficient or optimal choice. Bowen, who
pointed this out in 1943 ("The Interpretation of Voting in the Allocation of Economic Resources,"
Quarterly Journal of Economics, vol. 58, 27-48), thought that symmetrical distributions of
preferences were likely to be common for a number of public goods and that voting outcomes
may often, therefore, give reasonable approximations to optimal outcomes.
To get this result, it may not be necessary to assume that for each public good communities are
faced with a potentially large array of pairwise choices; in fact, this is not a procedure that is used
very widely. Instead, elected representatives will often decide upon the chosen amount. In this
circumstance, we can call upon another theorem from political economy: that elected
representatives act in ways that maximize their chances of re-election. If this is true, then no
elected representative is going to want to support amounts of the public good that don’t have a
high degree of community support. This will tend to lead to support for the preferences of the
median person, so that nobody feels too upset. The dynamics of desiring re-election will, as a
general matter, lead the choices of elected officials to cluster together near the centre of the
spectrum of possibilities.
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