Chapter 27
Public Goods
A public good is a good that can be consumed by more than one individual at a time, while a
private good is a good that can be consumed by only a single individual.1 When I take out my
lunch sandwich, I can take a bite or I can let you take a bite – but there is no way that both of us
can take the same bite (unless we want to think of some really gross scenarios). The sandwich bite
is what economists call “rivalrous” – and this rivalry is what characterizes private goods. When I
launch some fireworks out of my backyard, on the other hand, both you and I can enjoy the same
fireworks display without either of us taking away from the enjoyment of the other. The fireworks
display is therefore what economists call “non-rivalrous” – and this non-rivalry is what characterizes
public goods. As we will see, this gives rise to particular kinds of externalities – because I might
not consider the benefits you get from my fireworks as I decide how big to make them. In our
discussion of public goods, we therefore return to a topic we partially covered in Chapter 21, but
we do so now with the benefit of some game theory tools from Chapter 24.
While we will often consider the extreme cases of non-rivalry and rivalry, we should start by
pointing out that it is actually more appropriate to think of goods as lying somewhere on a continuum between complete rivalry and complete non-rivalry. Complete non-rivalry would mean that
we can keep adding additional consumers, and no matter how many we add, each new consumer
can enjoy the same level of the good without taking away from the enjoyment of others. National
defense is a good example of such an extreme: The national defense system of the U.S. protects
the entire population, and as new immigrants join the population or as new citizens are born, these
additional “consumers” can enjoy the same level of protection that current citizens enjoy without
making current citizens less safe from external threats. But if my city’s population increases, we
will need to get more police officers to keep public safety constant – which means that local public
safety is not as non-rivalrous as national defense. Or you and I can probably enjoy the same large
swimming pool without taking away from each other’s enjoyment, but as more people join, things
will get “crowded” and our enjoyment falls when new consumers come on board. Even my TV in
my living room is non-rivalrous to some extent, but my living room gets crowded even more quickly
than our local swimming pool.
The degree of non-rivalry then characterizes the degree to which we think of a good as being a
public good. My sandwich bite is on one extreme end of the spectrum – with even one other person
1 This chapter employs basic game theory concepts from Section A of Chapter 24 and refers frequently to our
analysis of externalities in Chapter 21. Chapters 25 and 26 are not required for this chapter.
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Chapter 27. Public Goods
Excludable
Non-Excludable
Types of Goods
Rivalrous (Private)
Non-Rivalrous (Public)
(Pure) Private Good
Club Good
Common (Private) Good Public (or Local Public) Good
Table 27.1: Different Kinds of Public and Private Goods
crowding my consumption to a point where it is no longer meaningful. National defense might be
on the other extreme – with no limit to the number of people that can be protected by the same
national security umbrella without “crowding” the protection enjoyed by everyone else. And then
there are all the in between goods – goods that can be consumed by more than one person at a
time but that are subject to crowding in the sense that, at least at some point, each individual’s
enjoyment of the public good falls when more people consume it. Within the class of public goods,
there are of course those that are quite local – like my TV or my local swimming pool – and
some that allow consumption over a wider geographic area – like national defense or reductions in
greenhouse gas emissions. The former are sometimes referred to as local public goods – and these,
like local public safety, in turn are typically (though not always) subject to some crowding within
the area in which they are provided.
While the degree of rivalry of a good is thus one dimension along which we can distinguish
between different goods (and the geographic reach of non-rivalrous goods is another), it will furthermore become important for us to distinguish between goods based on whether or not we can
exclude others from consuming the good. If you are my neighbor, I can’t exclude you from enjoying
my fireworks (unless I clobber you over the head and knock you unconscious), but I can exclude you
from my living room and thus from watching my TV. This will play an obvious role in how public
goods can be provided: If exclusion is possible, it is in principle (and often in practice) the case
that firms can charge consumers for their consumption of public goods and consumers can decide,
much as they do for private goods, whether it’s worth it to pay the price of admission. But if the
good is non-excludable, that option is not typically open to us. Firms are therefore much more
likely to provide excludable public goods than they are to provide non-excludeable public goods.
Table 27.1 illustrates four stylized types of goods that emerge from distinguishing goods along
the dimensions of rivalry and excludability. So far, we have almost always assumed that goods are
rivalrous – and thus we have dealt almost exclusively with private goods from the first column of the
table. Usually the private goods we have dealt with were excludable – with consumers who were
not willing to pay for such goods priced out of the market. In Chapter 21, however, we discussed
the case of private (rivalrous) goods to which multiple people have access. Such goods included
wood in a public forrest or fish in the ocean – goods not owned by anyone, goods that are part of
the “commons”. And we illustrated that lack of ownership (or “property rights”) of such private
goods results in the “Tragedy of the Commons” where individuals over-use the private good as they
do not consider the impact their actions have on others who also wish to make use of the good.
Over-consumption then resulted from the non-excludability of private goods in the “commons”.
We now turn to the second column in the table – public goods that are (at least to some extent)
non-rivalrous. When consumers cannot easily be excluded from consumption of such public goods
(as in the case national defense or my backyard fireworks), we will call them simply “public goods”
or, if their consumption is limited to small geographic areas, “local public goods”. Such public
goods might be “pure” in the sense that new consumers can always engage in consumption without
27A. Public Goods and their Externalities
1061
taking away from the consumption of current consumers (i.e. national defense and fireworks) or
they can be “crowded” (i.e. public safety in cities and public swimming pools). When there exists
a mechanism for excluding consumers (such as the case of the swimming pool or my TV), we will
sometimes refer to such goods as “club” goods. Again, the real world is much richer than this table
suggests because there are many cases in between the extremes, but this categorization will become
useful as we think about different ways in which goods can be provided by markets, governments
and civil society.
27A
Public Goods and their Externalities
We will begin with the case of fully non-rivalrous goods in the absence of excludability – or what
we just referred to as “pure” public goods in Table 27.1. In Section 27A.1, we will illustrate the
conditions that would have to be met in order for such public goods to be produced in optimal
quantities. We will see that decentralized behavior by individuals results in a fundamental externality problem – known as the “free rider problem” – that keeps individuals on their own from
providing optimal quantities of the public good. And we will see that this fundamental problem
is yet another incarnation of the Prisoners’ Dilemma. Put differently, for the case of such “pure”
public goods, the first welfare theorem does not hold – decentralized individual behavior does not
result in optimal outcomes – because of the strategic considerations that guide individual behavior
in the presence of externalities.
For the remainder of part A of this chapter, we will then investigate different approaches for
solving this free-rider problem. The classic solution is to look toward government intervention
which we will investigate in Section 27A.2. In Section 27A.3 we then ask, given our understanding
of externalities as a problem of “missing markets”, to what extent market forces could assist in the
provision of some types of public goods, in particular those that are excludable (which we referred
to as “club goods” in Table 27.1) and those that are local. In the process we will identify a second
fundamental problem that plagues both government and market solutions to the free rider problem:
the problem that individuals often have an incentive to mis-represent their tastes for public goods.
In Section 27A.4, we discuss a possible role of civil society institutions and, in the process, we will
refer back to the Coase Theorem from Chapter 21 while also thinking of how individuals might
partially overcome the free rider problem through the evolution of tastes that include a particular
taste for giving. Finally, we will return to the problem of the incentive to misrepresent tastes for
public goods in Section 27A.5 and will ask to what extent it might be possible for government or
private institutions to overcome this problem through the clever design of incentive mechanisms
that make it in people’s best interest to tell the truth.
27A.1
Public Goods and the Free Rider Problem
In panel (a) of Graph 27.1, we begin by replicating panel (a) from Graph 14.1 in Chapter 14. In
that graph, we had illustrated how we add up individual demand curves in the case of a private
good. Since private goods are rivalrous and can be consumed by only one person, this addition of
demand curves was “horizontal” in nature – for every additional consumer, we simply added that
consumer’s demand at each price level to the previous demand curves. Public goods are different
because they are non-rivalrous – that is, they can be consumed by more than one person at a time.
Thus, in order to derive the aggregate marginal willingness to pay for 1 unit of the public good,
we have to add how much that good is worth to the first consumer to how much it is worth to
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Chapter 27. Public Goods
the second consumer and so forth. When tastes are quasilinear, we can equivalently say that this
amounts to adding demand curves “vertically”. This is done in panel (b) of Graph 27.1.
Graph 27.1: Aggregate Demand Curves for Private and Public Goods
27A.1.1
The Optimal Level of Public Goods
Now suppose that the good on the horizontal axis can be produced at constant marginal cost. In
the private good case, the efficient level of production then occurs where marginal cost intersects the
aggregate (or “market”) demand curve DM in panel (a) of Graph 27.1 (as we showed in Chapter 15).
At that intersection point, it was then the case that each consumer’s marginal willingness to pay
was equal to the marginal cost of production, and when the private good represented a composite
good denominated in dollar units, this is equivalent to saying that each consumer’s marginal rate
of substitution (M RS) was equal to the marginal cost of production.
Now consider a public good that can similarly be produced at constant marginal cost. It is still
the case that efficiency requires that the good be produced so long as the marginal benefit of the
good is greater than the marginal cost, but now all the consumers who consume the same public
good are receiving a marginal benefit from doing so. To say that the efficient level of production
of the public good occurs where marginal benefit is equal to marginal cost is therefore the same
as saying that production occurs where the sum of the marginal benefits of all consumers equals
the marginal cost. In a sense, exactly the same is true in the private goods case, except there the
sum of the marginal benefits is only the marginal benefit of a single consumer since no good can
be consumed by more than one person.
Exercise 27A.1 True or False: The efficient level of public good production therefore occurs where marginal
cost crosses the aggregate demand for public goods as drawn in Graph 27.1b.
Exercise 27A.2 * Can you explain how there is a single efficient level of the public good when tastes for
public goods are quasilinear – but there are multiple levels of efficient public good provision when this is not
27A. Public Goods and their Externalities
1063
the case? (Hint: Consider how redistributing income (in a lump sum way) affects demand in one case but
not the other.)
There is another way we can derive this optimality condition for public good production. Remember that a situation is “(Pareto) optimal” or “efficient” if there is no way to change the situation
and make some people better off without making anyone else worse off. Suppose then that we consider the case of two consumers with preferences over a composite private good x and a public
good y and with private good endowments e1 and e2 . Suppose further that there exists a concave
production technology that converts private goods x into public goods y. We can then depict the
tradeoffs that our “society” of two individuals faces with the green “production possibilities frontier” in panel (a) of Graph 27.2 where the two consumers could have only private consumption
(equal to e1 + e2 ) on the vertical axis, or they could devote some of their private goods to producing a public good that they can both consume. A concave production technology implies that
relatively little private good is needed to produce the first units of the public good but that it takes
increasingly more private goods to produce each additional unit of the public good. As a result, the
tradeoff that emerges takes on the shape depicted in the graph, with an initially shallow slope that
becomes increasingly steep as more public goods are produced. The slope of this graph represents
the number of x units required to produce one more unit of y – or the (negative) marginal cost
(−M Cy ) in terms of x goods – for producing another unit of public good.
Exercise 27A.3 Does this production technology exhibit increasing or decreasing returns to scale?
Exercise 27A.4 What would the relationship in the graph look like if the technology had the opposite returns
to scale as what you just concluded?
In panel (b) of the graph, we then pick some (magenta) indifference curve for consumer 2 and
place it onto the graph of the production possibilities frontier. The slope of an indifference curve is
the marginal rate of substitution, or put differently, the amount of x consumer 2 would be willing
to give up in order to get one more unit of y. Another way of expressing this is that the slope of
the indifference curve is simply minus consumer 2’s marginal benefit (−M B2 ) of one more unit of
y expressed in terms of x.
Now let’s see how high an indifference curve we could get for consumer 1 assuming we make
consumer 2 no worse off than the indifference curve u2 . If we were to produce y in panel (b) of
the graph, we would have to give all remaining x goods to consumer 2 just to keep her at the
indifference curve u2 – leaving us no x goods to give to consumer 1. The same is true were we to
produce y. But for public good levels in between y and y, we would have some x goods left over
to give to consumer 1. Panel (c) of Graph 27.2 then plots the amount of x that is left over for
consumer 1 for each level of y good production between y and y.
Exercise 27A.5 Why must the shaded areas in panels (b) and (c) of Graph 27.2 be equal to one another?
It is now easy to see in panel (c) of the graph how high an indifference curve for consumer 1
we can attain assuming consumer 2 is held to indifference curve u2 . All we have to do is find the
highest indifference curve for consumer 1 that still contains at least one point of the shaded set of
possible (x, y) levels we have derived – leading to a public good level y ∗ at which the indifference
curve u∗1 is tangent to the boundary of the shaded set in panel (c). This boundary of the shaded
set is simply the production possibility frontier minus the indifference curve u2 , which implies that
the slope of the boundary of the shaded set is the difference between the slopes of the production
possibilities frontier and the indifference curve u2 – i.e. −M Cy − (−M B2 ) = −M Cy + M B2 . At the
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Chapter 27. Public Goods
Graph 27.2: Optimal Provision of Public Goods
tangency that occurs when public goods are set at y ∗ , this slope equals the slope of the indifference
curve u∗1 , which implies that −M B1 = −M Cy + M B2 . Subtracting M Cy from both sides of this
equation and adding M B1 , we therefore get that M B1 + M B2 = M Cy .
The only thing that seems arbitrary about what we just did is that we just picked some indifference curve for consumer 2. But notice that the reasoning does not depend on what indifference
curve for consumer 2 we pick in panel (b) as long as some shaded area remains. Thus, no matter
what feasible indifference curve for consumer 2 we choose, finding the public good level that insures
we cannot make consumer 1 better off without making consumer 2 worse off implies picking y such
that M B1 + M B2 = M Cy . Thus, of the many possible (Pareto) optimal solutions we can think of
(as we vary u2 ), all of them share in common that the public good level is set so that the sum of
marginal benefits of the public good equals the marginal cost of producing public good. This is in
contrast with the efficiency condition for private goods where (assuming all consumers are at an
interior solution) each individual M Bi equals the marginal cost.
Exercise 27A.6 * Is there any reason to think that y ∗ – the optimal level of the public good – will be the
same regardless of what indifference curve for consumer 2 we choose to start with? How does your answer
27A. Public Goods and their Externalities
1065
change when tastes are quasilinear in the public good? And how to does this relate to your answer to exercise
27A.2?
27A.1.2
Decentralized Provision of Fireworks
Suppose now that we consider a particular example. A national holiday is approaching, and you
and I are planning to celebrate by launching fireworks in our backyards. The resulting fireworks
are a public good – my enjoyment as I glance up into the evening sky does not take away from your
enjoyment, and I will get to enjoy the fireworks you launch just as you will enjoy the ones launched
from my backyard. We should probably get together and pool our resources in order to arrive at
the Pareto opitmal level of fireworks y ∗ , which, as we just derived, implies that y ∗ would be set
such that the sum of our marginal benefits equals the marginal cost of launching an additional
firework. But instead, we go about our business and determine the number of fireworks we launch
independently of one another knowing that the other is also doing so.
To estimate how many fireworks will be launched by each one of us, we then have to figure out
the Nash equilibrium of the game we are playing as we try to anticipate how many fireworks the
other will launch. In a Nash equilibrium, my level of firework production must be a best response
to your level of firework production and vice versa. We therefore begin by thinking about my best
response to any quantity of fireworks you might launch.
If I thought you were not going to launch any fireworks (i.e. y2 = 0), I would invest in my own
fireworks until the marginal cost of launching one more firework is equal to the marginal benefit I
receive – i.e. I will set y1 (0) such that M B1 = M C. If I think you will produce some quantity y 2 , I
will have to re-think how many fireworks I will launch because I know I already get to enjoy y 2 > 0
of your fireworks. You purchasing fireworks is a lot like me having additional disposable income –
because I could now simply enjoy your fireworks and spend all my income on private goods. If all
goods are normal goods, the additional income I now have will be split between all goods, which
means I will not spend all the effective additional income on the public good. Put differently, while
I will end up consuming more fireworks if you buy some, I will purchase less myself.
Exercise 27A.7 In a graph with y on the horizontal axis and a composite private good x on the vertical,
illustrate my budget constraint assuming that y 2 = 0. How does this budget constraint change when y2 > 0?
Show that, if tastes are homothetic, I will end up consuming more y when y 2 > 0 but will myself purchase
less y. Does this hold whenever y and x are both normal goods? Does it hold if y is an inferior good?
In panel (a) of Graph 27.3, we can then illustrate my best response function to different values
of y2 that you might choose on a graph with y2 on the horizontal axis and y1 on the vertical. Our
reasoning above implies that this best response function has a positive intercept y1 (0) when y2 = 0
(i.e. I will purchase fireworks until M B1 = M C) but negative slope (i.e. as y2 increases, I buy
fewer fireworks.) In panel (b), we put your best response function on top of mine assuming that
you are just like me – with the two best response functions therefore crossing on the 45-degree line.
That intersection then represents the levels of fireworks (y1eq , y2eq ) that we will buy in equilibrium
when we both best respond to the other’s actions.
Exercise 27A.8 If you and I have identical tastes but I have more income than you, would the equilibrium
fall above, on or below the 45 degree line (assuming all goods are normal goods)?
We can now ask if the total quantity of fireworks y eq = y1eq + y2eq is efficient. In equilibrium, I am
doing the best I can if I continued to buy fireworks as long as, given that you are purchasing y2eq ,
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Chapter 27. Public Goods
Graph 27.3: Private Provision of Public Goods
my own marginal benefit of additional fireworks was greater than the M C; i.e. I would stop when
M B1 = M C. Since you also get a benefit from the fireworks I launch in my backyard, this implies
that I stop buying fireworks when M B1 +M B2 > M C – which implies that the equilibrium quantity
of fireworks is less than the efficient quantity y ∗ for which we concluded before M B1 + M B2 = M C.
Thus, y eq < y ∗ – in equilibrium we are producing an inefficiently low quantity of fireworks.
The intuition for the result is straightforward and easy to understand given our work on externalities Chapter 21. When I make my choice on how many fireworks to buy, I am generating a
positive externality for you but I have no incentive to take that into account. The same is true for
you. Because we have no incentive to take into account the benefits we are producing for others,
we will under-consume fireworks. This is often referred to as the free rider problem – each of us is
“free-riding” on the public good produced by the other.
27A.1.3
The Free Rider Problem: Another Prisoners’ Dilemma
This free rider problem is yet another example of a Prisoners’ Dilemma. You and I could, after all,
have gotten together before going to the fireworks store and agreed to split the cost of buying the
optimal quantity of fireworks. Instead, we acted independently and did not explicitly cooperate.
But even if we had chosen to coordinate beforehand and had agreed to each buy our share of
the optimal quantity of fireworks, we would not have had an incentive to actually abide by our
agreement regardless of what we thought the other was doing. This is because our private incentive
is to behave in accordance with our best response functions in Graph 27.3, setting our private
marginal benefit equal to the marginal cost we incur. Thus, in order to sustain cooperation when
we get to the store, we need a mechanism to enforce our agreement. Our incentives are exactly like
those of the oligopolists who make a cartel agreement in Chapter 25 – abiding by the agreement
would in fact make both of us better off than we are by going at it alone, but, if there is no one to
make sure we actually abide by the agreement, it is in our individual incentive to cheat.
In our fireworks example, we might easily be able to imagine that we could in fact think of an
27A. Public Goods and their Externalities
1067
enforcement mechanism. All we have to do is have one of us buy the optimal number of fireworks,
have the other pay half the bill and then get together in one of our backyards and blast off all
the fireworks. Even in the absence of being so explicit about enforcing our agreement, we might
think it’s enough for us to know that we are likely to be neighbors for a long time and that we will
keep having occasions to cooperate on the fireworks we launch. As we have seen in Chapter 24,
introducing the likelihood that we will interact repeatedly (without knowing a definitive end to the
game) can in fact be enough for us to sustain cooperation in repeated interactions. We will think
a bit more about circumstances under which we think private actors are likely to find ways out of
the prisoners’ dilemma in Section 27A.4.
More generally, however, there are many circumstances involving public goods where it is unlikely that it will be so easy to figure out ways of overcoming the incentives of the one-shot prisoners’
dilemma. Many public goods involve many players, and it is difficult for large numbers of players
to cooperate the way that you and I might when we prepare for our fireworks. Not only is it more
difficult to enforce cooperation, but the incentives to free ride on the contributions of others get
worse the more “others” there are. (You can explore this further in end-of-chapter exercise 27.1.)
We all benefit from investments in cancer research, but the American Cancer Association cannot
easily get us all to consider the larger social benefits of cancer research when it appeals to individuals to contribute to the cause. We all benefit from an effective police force that keeps us relatively
safe, but it’s not easy to see how the police can simply walk around and collect the optimal level
of donations for its worthwhile work. For this reason, we often look to non-market institutions like
governments to bring our private incentives in line with socially desirable levels of investments in
public goods.
27A.2
Solving the Prisoners’ Dilemma Through Government Policy
As we have already seen in previous chapters, governments are often employed as non-market
institutions that enforce ways out of prisoners’ dilemmas. There are at least two possible avenues
for governments to do so: First, in many cases governments simply take on the responsibility of
providing public goods and use the power to tax individuals to finance those goods. Second, in some
cases governments do not directly provide public goods but instead subsidize private consumption
of public goods. Each can, assuming governments have sufficient information, result in optimal
levels of public good provision.
27A.2.1
Government Provision and “Crowd-Out”
Perhaps the most straightforward solution to the public goods/free rider problem is for the government to simply provide the public good directly. This happens in most countries for goods such as
national defense or the establishment of an internal police force. But the argument for government
provision of public goods has also been used to justify income redistribution programs in most
Western democracies where it is assumed that most citizens place some value on making sure the
least well off are taken care of to some extent. Assuming that this is the case, contributions to the
alleviation of poverty are in fact contributions to a public good because everyone who cares about
the issue benefits from less poverty.2
2 End-of-chapter exercise 27.8 explores this argument in some more detail. Of course an alternative explanation for
the existence of redistributive programs arises from a desire by voters to establish insurance markets when private
markets are missing due to adverse selection. We discussed this in Chapter 22 for cases such as unemployment
insurance.
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Chapter 27. Public Goods
When governments do not know exactly what the optimal level of a particular good is, or alternatively, if political processes are not efficient and therefore do not result in optimal economic
decision making, a particular issue called “crowd-out” may arise. Consider, for instance, government financing for public radio. In the U.S., the federal government in fact finances part of the
cost of operating public radio stations, but radio stations attempt to get listeners to add private
contributions on top of the funds received from the government. The government is, as a result,
just one of many contributors to the provision of the public good “public radio”, and public radio
listeners will presumably think about their own level of voluntary contribution in light of how much
others are giving – with “others” including the government’s contribution.
The resulting “game” is then not at all unlike the game in which you and I are trying to decide
how much to contribute to our local fireworks except that now there is just another player called
the government. We derived in the previous section an individual’s best response function in such
a game as a function of how much others are giving to the public good, and we noticed that as
others give more, each individual’s best response is to give less. When the government therefore
contributes to a public good (such as public radio) that also relies on private contributions, game
theory predicts that private contributions will decline as government contributions increase – or, to
use the economist’s language – government contributions to the public good “crowd out” private
contributions. In fact, as we will see more formally in Section B, if the government taxes individuals
in order to finance its contribution to a public good, the model would predict that individuals
who are giving to the public good will reduce their contributions by exactly the amount that the
government has taken from them in order to finance the same public good. Thus, so long as
individuals are giving on their own, we would expect increased government contributions to exactly
offset decreases in private contributions.
Exercise 27A.9 True or False: If everyone is currently giving to a public good – including the government –
then this model would predict that the government’s involvement has not done anything to alleviate the
inefficiency of private provision of public goods.
In the case of public radio, or course, not every tax payer is also giving voluntarily to public radio
stations. The tax revenues raised for public radio from individuals who are not giving therefore
do not result in decreased private contributions since those individuals are already at a “corner
solution” where they do not give anything to public radio. In part for this reason, we do not
see government contributions to public goods in the real world accompanied by dollar for dollar
decreases in private contributions. In the case of public radio, it appears that an increase of $1 in
government contributions is accompanied by a decrease in the range of 10 to 20 cents in private
contributions.3
Exercise 27A.10 Could it be that an increase of government support for a public good causes someone
who previously chose to give to that public good to cease giving? How would such a person’s best response
function look?
27A.2.2
Government Provision under Distortionary Taxes
Another real world problem governments face is that, as we have emphasized earlier in this book,
governments are rarely able to use non-distortionary taxes to raise revenues. If a government does
3 See Kingma, Bruce (1989),“An Accurate Measurement of the Crowd-out Effect, Income Effect, and Price Effect
for Charitable Contributions,” Journal of Political Economy 97(5), 1197-1207.
27A. Public Goods and their Externalities
1069
find a non-distortionary or efficient tax (that generates no deadweight loss), it would in fact be
optimal for it to provide the public good level y ∗ at which the sum of individual marginal benefits
is equal to the marginal cost of providing the public good. But if distortionary taxes have to be used
in order to raise revenues for public good provision, the social marginal cost of government provision
is higher than simply the cost of producing the public good because each dollar in tax revenues
raised is accompanied by a deadweight loss. Thus, the optimal level of government provided public
goods decreases the more distortionary the taxes used to finance public goods are. (This is explored
further in end-of-chapter exercise 27.9.)
Exercise 27A.11 Given what we have learned about the rate at which deadweight loss increases as tax rates
rise, what would you expect to happen to the optimal level of government provision of a particular public
good as the number of public goods financed by government increases?
Exercise 27A.12 If a particular public good is subject to some partial “crowd-out” when governments
contribute to its provision, might it be optimal for the government not to contribute to the public good in
the presence of distortionary taxation?
27A.2.3
Subsidies for Voluntary Giving
An alternative policy to government provision of a public good involves the government subsidizing
the private production of the good. This, too, should be intuitive as soon as we recognize the free
rider problem as arising from the presence of a positive externality. In our Chapter 21 treatment of
externalities, we in fact illustrated that the under-provision of goods due to positive externalities
can be corrected through what we called Pigovian subsidies.
Suppose, for instance, that our local city government finds it just silly that you and I keep
falling victim to prisoners’ dilemma incentives when we put up our annual fireworks display. So the
government decides to make it cheaper for each of us to buy fireworks by paying for some portion
s of each firework we purchase. You and I will still be playing the same game we did before, except
that our best response functions will now shift up. Remember that my best response to any public
good level y2 that you purchase is determined by the condition that my marginal benefit from the
last unit of public good I purchase will be equal to the marginal cost of making the purchase. If
the government pays for a portion of each firework I buy, my marginal cost falls – which implies I
will purchase more fireworks for any expectation I have of y2 than I did before. Graph 27.4 then
illustrates how both of our best response functions (and thus the Nash equilibrium) change as the
subsidy increases from panel (a) through (c). In panel (a), we have no subsidy where we each
purchase less than half the efficient quantity y ∗ . In panel (b), a modest subsidy shifts our purchases
closer to the equilibrium, and in panel (c) the subsidy is exactly the size it needs to be in order for
both of us to purchase half the efficient quantity (and together we therefore purchase y ∗ ).
Exercise 27A.13 In Section B we show mathematically that the optimal subsidy will involve the government
paying for half the cost of the fireworks if you and I have the same preferences. By thinking about the size
of the externality – i.e. how much of the total benefit that is not taken into account by an individual
consumer — does this make intuitive sense?
Exercise 27A.14 Could the government induce production of the efficient level of fireworks if it only
subsidized the purchases of one of the consumers?
In the real world, the most common way in which governments fund private giving for public
goods is through tax deductions. The U.S. income tax code, for instance, allows individuals to
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Graph 27.4: The Changing Nash Equilibrium under Subsidies
give to charitable institutions and not pay taxes on the amount that they give to such institutions.
Thus, if I give $100 of my income to the American Cancer Society, I get to deduct this from the
income on which I would otherwise have to pay taxes. If my marginal income tax rate is 30%, I
then have a choice of either paying $30 of the $100 in income taxes and spending the remaining $70
on stuff I like to consume, or I can give $100 to the American Cancer Society. Giving $100 to the
American Cancer Society therefore costs me only $70 in private consumption. Thus, by making my
charitable contribution tax deductible, the government has subsidized my contributions by 30%.
Exercise 27A.15 Under an income tax that has increasing marginal tax rates as income goes up, the rich
get a bigger per-dollar subsidy for charitable giving than the poor when charitable giving is tax deductible.
Americans make heavy use of this subsidy for giving to charitable organizations that, at least
to some extent, provide public goods. Organizations that receive such subsidized contributions
include churches, hospitals, organizations (like the American Cancer Society) that fund research,
art galleries, museums, etc. Chances are, if you are taking this course in an American university,
your university has received substantial private contributions from individuals that deduct these
contributions from their income taxes, and your university is providing public goods such as contributing to the creation of knowledge through the research activities supported by the faculty at
your university.
Exercise 27A.16 If the only way to finance the subsidy for private giving is through distortionary taxation,
would you expect the optimal subsidy to be larger or smaller than if the subsidy can be finance through efficient
lump sum taxes?
27A.3
Solving the Prisoners’ Dilemma By Establishing Markets
In Chapter 21, we saw that, at a fundamental level, the “market failure” that arises from the
existence of externalities is really a “failure of markets to exist”. And we argued that, hypothetically,
if sufficient numbers of markets were established, the externality would disappear and with its
disappearance, the first welfare theorem would reappear. We will therefore investigate next the
extent to which we can think of markets as a possible solution to the public goods problem.
27A. Public Goods and their Externalities
1071
We could apply this at a purely abstract level to our fireworks example. The fundamental public
goods (and free rider) problem emerges from the fact that, when I consume fireworks, I am also
producing fireworks consumption for you. But there is no market that prices the production of
fireworks consumption for you – i.e. there is no price that you have to pay me when I produce
something that you value. As a result, I do not take into consideration the benefit that you incur
from my fireworks. There is a positive externality, which is the same as saying there is a missing
market for goods that are being produced as I make my consumption decision. It is not at all clear
how we would establish the missing market for my fireworks production, nor would there be much
of a “market” with only two of us involved. The point is therefore not to argue that such markets
could generally be established. But neither does the difficulty of establishing the abstract “missing
markets” mean – as we saw in the example of negative pollution externalities and pollution voucher
markets – that we cannot consider some form of market solution to the problem.
We therefore want to think about the conditions under which decentralized market provision of
public goods could emerge if certain types of markets were appropriately set up. In order for us to
have any chance of public goods being provided in such a decentralized market setting, it would seem
that at the very least we have to assume that consumption of the public good is excludable; that is
we would have to assume that the producer of the public good can keep people from consuming the
good if they do not pay what the producer demands. This does not take away from the non-rivalry
of the good – that is, the public good can still be consumed my multiple people at the same time.
For instance, a large swimming pool can be enjoyed by a large number of families at the same time,
but the provider of the swimming pool can keep people out if they don’t pay an entrance fee.
Exercise 27A.17 Can you think of other goods that are non-rivalrous (at least to some extent) but also
excludable?
27A.3.1
Lindahl Price Discrimination and the Incentive to Lie
Decentralized market exchanges are governed by prices, and in our typical competitive equilibrium,
this meant that everyone faces the same market price and each consumer gets to choose her optimal
quantity at that price. Same price, different quantities. Now let’s ask how a “market” for a typical
pure public good would have to look. A pure public good is a good that all consumers can consume
at the very same time in the same quantity. So in a “market” for public goods, individuals would
consume the same quantity of the public good. But, in order for that quantity to be something
the consumer actually chooses given her budget constraint, different consumers would have to face
different prices. Different prices, same quantity – the exact opposite of the decentralized market
equilibrium for private goods.
Consider the case of fireworks and suppose that a producer of fireworks displays owns a sufficiently large land area such that the only way to see the fireworks is to actually step onto the
producer’s land. Suppose further that the producer has put up barbed wire around his land with,
just to be mean, a sufficiently strong electrical current flowing through the wire to instantly knock
any potential trespasser unconscious. The only way to step onto the land is to go through an
entrance booth at which the producer can charge individuals an entry fee.
Now suppose the producer knows each consumer’s demand curve for the intensity of firework
displays – and we can thus determine the optimal number of fireworks y ∗ to launch into the air
during a particular holiday. Recall that we can calculate y ∗ by simply adding the demands vertically
and finding where the resulting aggregate demand curve intersects marginal cost. Our producer
of fireworks can then determine individualized prices for each consumer such that each consumer
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would in fact choose y ∗ as part of her optimal consumption bundle at her own individual price.
The individualized price for consumer i would then simply be her marginal benefit of the public
good y ∗ , and since the marginal benefits sum to marginal cost at y ∗ , the individualized prices sum
to marginal cost.
Exercise 27A.18 Illustrate, using a graph of two different demand curves for two different consumers, how
a producer would calculate y ∗ – and what prices she would charge to each individual in order to get them to
in fact choose y ∗ as their most preferred bundle.
Exercise 27A.19 Does the producer collect enough revenues under such individualized pricing to cover
marginal costs?
The resulting equilibrium would be one in which a single producer of the public good charges
different prices to consumers in such a way that each consumer chooses the same quantity of the
public good. This is the public good analog to the private good competitive equilibrium, and it is
known as a Lindahl equilibrium.4 The prices that emerge in this equilibrium are known as Lindahl
prices. Note that it involves price discrimination by the producer, with higher prices charged to
consumers that have greater demand for the public good. But in order to implement the price
discrimination, the producer has to know the demands (or preferences) of individual consumers.
And therein lies the problem with the Lindahl equilibrium.
Since I know that the price I will be charged as I enter the land on which I can view the fireworks
is directly related to the producer’s impression of my tastes for fireworks, I have every incentive to
play down how much I actually like fireworks. “I can’t believe I am going to see another stupid
fireworks display,” I will mutter on my way toward the gate – just loud enough for the fireworks
producer to hear me. Put differently, I have an incentive to lie about my preferences. And, what’s
worse, that incentive increases the more people are lining up to get onto the land from which the
fireworks can be enjoyed. If you and I are the only ones to see the fireworks, I face a trade-off
when I decide on how much to lie about my enthusiasm for fireworks: On the one hand, any lie will
reduce the number of fireworks that will be launched (because it will affect the calculation of y ∗ ),
but, on the other hand, I will not have to pay as much to get in if I lie. So I’ll lie a little bit but
won’t claim that I don’t care about fireworks at all. If, however, there are 10,000 people lined up
to get onto the land from which the fireworks display can be enjoyed, I am suddenly only one of
many. This means that the impact of my lie on y ∗ becomes very small, but the impact of my lie on
the price I’ll get charged continues to be big. As the number of consumers goes up, the incentive
to lie therefore increases because the impact of a lie on y ∗ diminishes with more consumers but the
impact of the lie on the price I get charged does not. Unless producers of public goods already know
a lot about the preferences of their consumers, a Lindahl equilibrium under which consumers choose
the optimal quantity of the public good at individualized prices therefore cannot emerge because
the consumers have a strong incentive to mis-represent their preferences for the public good.
Exercise 27A.20 Consider the entrance fees to movie theaters on days when not every seat in the movie
theater fills up. If it is generally true that older people and students have lower demand for watching new
releases in movie theaters, can you explain entrance discounts for the elderly and for students as an attempt
at Lindahl pricing?
One could argue that private goods markets also face such incentive problems – that, when you
and I negotiate over the price I will pay you for a gallon of milk, I also have an incentive to pretend
4 This
is named after Erik Lindahl (1891-1960), a Swedish economics, who first proposed the idea in 1919.
27A. Public Goods and their Externalities
1073
that the milk is not worth that much to me so that you’ll give it to me at a lower price. That’s
true – but the difference is that my incentive to lie about my tastes for milk get weaker and weaker
the more milk consumers there are because if I claim to not like milk that much, you’ll just go to
someone that isn’t such a pain. Thus, in private goods markets the incentive to misrepresent our
preferences disappears as the market becomes large, while in public goods markets that incentive
gets bigger and bigger the larger the market. I doubt it has ever even occurred to you to try to tell
the local supermarket owner that you really don’t care for milk that much in order to get a better
price, but if I came to you and told you that your taxes will increase the more you tell me you like
national defense but the increased tax payments from you will have no perceptible impact on the
level of national defense, you’d probably pretend to be a pacifist singing “Give Peace a Chance”
pretty quickly.5
27A.3.2
“Clubs,” Local Public Goods Markets and “Voting with Feet”
The concept of a Lindahl Equilibrium, while academically interesting, is therefore of limited realworld usefulness given the necessity for producers to know consumer preferences that consumers
themselves have every incentive to misrepresent. That does not, however, mean that other forms
of market forces might not play an important role in shaping the kinds and varieties of excludable
public goods we can choose. Homeowners’ Associations offer public security, swimming pools and
golf courses; a variety of “clubs” offer access to public spaces to paying customers; and local
governments of all kinds offer a variety of public services. The goods offered by such institutions
are not “pure” public goods that are fully “non-rivalrous”, but each can still be consumed by
multiple consumers at the same time. And in each case, market forces play an important role.
This was pointed out by Charles Tiebout (1924-68) in the 1950’s and has given rise to one
of the largest academic literatures in all of economics.6 Tiebout proposed a simple and intuitive
hypothesis: When there are goods that are neither fully rivalrous nor fully non-rivalrous, and
when there exists a mechanism for excluding consumers who do not pay the required fee for using
the good, one can derive conditions under which multiple providers of such goods will compete
in a market-like setting and provide efficient levels of the goods. Tiebout was thinking of local
communities as being the providers, with local public services restricted to those who reside within
the boundaries of local communities. Just as different malls and shopping centers provide different
varieties of stores and different levels of characteristics (such as lighting in parking lots, a private
security force to protect the mall, etc.) that consumers might care about, we can think of different
communities providing different mixes of public services with different mixes of local fees and taxes
for residents of those communities. Just as malls compete with one another for customers who
5 In Chapter 16, we argued that the concept of a competitive equilibrium becomes particularly compelling once
we realize that the set of stable allocations in the world – formalized in the concept of the “core” set of allocations –
converges to the set of competitive equilibrium allocations as an economy becomes large. It can be shown that the
opposite is true for public goods economies: as the economy becomes large, the set of core allocations explodes far
beyond just the allocations that could be supported in a Lindahl equilibrium. The reason for this is closely related
to the reason why the incentive to misrepresent one’s preferences increases as the economy gets large.
6 The argument was presented in a quite accessible article – see Tiebout, C. (1956) “A Pure Theory of Local
Expenditures,” Journal of Political Economy 64, 416-24 which has become one of the most cited articles in economics.
It was written while Tiebout was a graduate student at the University of Michigan. He died suddenly at a relatively
young age, and his relatives appear not to have realized the importance of his contributions. I know this from
personal experience: I once gave a paper at a university workshop and was afterwards approached by an elderly man
who told me he had no real idea what on earth I had been talking about in my 90 minute presentation – but he just
wondered whether my reference to the “Tiebout Model” in the title of my paper had anything to do with his “cousin
Charles”. Turns out it did.
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Chapter 27. Public Goods
will decide to frequent one mall or shopping center more than others, communities then compete
for residents. Successful malls find sufficient numbers of consumers with similar tastes to create a
sufficiently large clientele, as do successful communities.
To the extent to which there is enough competition between shopping centers, each center will
make roughly zero profits in equilibrium and consumers can choose from the optimal number of
different centers to find those that most closely match their tastes given their budgets. And to
the extent to which there is sufficient competition between local communities, such communities
similarly offer a variety of bundles of goods and services for consumers to choose from – with
each community’s choices disciplined by competitive market forces. In the case of communities,
land then serves as the exclusionary device since only those who own or rent land (and housing)
in a particular community have access to the public services offered. Such communities could be
privately operated (as are, for instance, homeowners’ associations) or publicly administered (as, for
instance, local school districts). And even when local governments are operated through political
processes, politicians have to confront market pressures to ensure that the mix of public services
and local taxes attracts a sufficient clientele of local residents.
Exercise 27A.21 Why do consumers not face the same incentive to lie about their tastes in such a
“Tiebout” equilibrium as they do in a Lindahl equilibrium?
Clubs that are not tied to land offer another application of Tiebout’s insight. One can think,
for instance, of churches as clubs providing public goods such as religious services, with churches
competing for parishoners who have different tastes for the types of music, sermons and denominational affiliations that are offered. While churches typically do not charge an entrance fee, they find
other ways of enforcing expectations about contributing to the church in financial and non-financial
ways (as we will discuss more a little later). Or one can think of private schools that offer a service
that has at least some public goods characteristics, with such schools competing on both the types
of curricula they offer and the level of tuition they charge. Or we can think of private operators of
swimming pools and health clubs who charge for uses of their somewhat non-rivalrous goods and
compete with others that do the same.
Exercise 27A.22 In recent years, gated communities that provide local security services privately have
emerged in many metropolitan areas that are growing quickly. Can you think of these from “club” perspective?
For a much richer treatment of these topics, you should consider taking a course on local public
finance or a course on urban economics where Tiebout’s insights are typically discussed at length.
As with many economic theories, the insights rarely hold perfectly in the real world but they do
play an important role in the bigger picture of how public goods are provided. For now, our main
point is just that, in speaking as if there is a crass distinction between “private goods” and “public
good”, we are implicitly ignoring a whole set of important goods that lie in between the extremes –
and the in-between cases are often provided by a rich combination of civil society, market and
government actions.
27A.3.3
The Lighthouse: Another Look at Excludability and Market Provision
In our discussion of market provision of public goods, we have placed some emphasis on the importance of “excludability” of public goods if such goods are to be provided through market forces.
After all, if a provider cannot exclude those who attempt to free-ride, how can she ever expect to
collect sufficient revenues to provide anything close to the optimal level of the public good?
27A. Public Goods and their Externalities
1075
There is much truth in the intuitive insight that providers (other than governments that can use
taxes) must find ways to finance public goods, and that this typically involves some mechanism for
excluding non-payers. But we sometimes underestimate the extent to which providers might find
creative ways of doing this. In a famous article, Ronald Coase studied the particularly revealing
case of lighthouses in the 18th century. Until Coase’s case study, the lighthouse was often given as a
motivating example in textbooks to illustrate the difficulty of providing a vital public good without
the government doing so directly. Before the invention of the current navigational technologies used
on ships, lighthouses played a pivotal role in guiding ships safely along dangerous shores where, in
the absence of the guidance offered by lighthouses, ships could easily run aground. The services
offered by lighthouses are classically non-rivalrous – no matter how many ships are safely guided
toward the shores by a lighthouse, additional ships can similarly make use of the light that is
emitted. And economists writing about the problem of providing lighthouses could not see an easy
way for private lighthouse operators to exclude those who do not pay.
Coase, however, looked to see how lighthouses were actually provided in many instances, and
what he found was that private providers had indeed found ways of financing lighthouses by charging
those who benefitted most from them. It turns out that providers bundled the public good provided
by the lighthouses with private goods – in particular the rights to dock a ship in the harbor to
which the lighthouses guided ships.7 While it is true that lighthouses offered additional positive
externalities to ships that simply used the light to navigate the shore without docking in the harbor,
it appears that these externalities were small relative to the benefits that could be priced for those
who used the local harbors. While the British government played a role in the protection of property
rights and the collection of light fees, it was not necessary to have the government directly provide
lighthouses.
Exercise 27A.23 Can you think of the provision of free access to swimming pools in condominium complexes in a way that is analogous to Coase’s findings about lighthouses?
27A.4
Civil Society and the Free Rider Problem
When we introduced the prisoners’ dilemma in Chapter 24, we pointed out that the model’s prediction of complete non-cooperation is often contradicted by experimental and real world evidence.
In the real world, people simply do not seem to free-ride nearly as much as our model predicts.
As a result, our model does not successfully predict the level of voluntary contributions to public
goods that we observe in the world. Nor does the model make sense of the distribution of charitable
giving – or, to be more precise, the model cannot make sense of the fact that the same person is
often observed to give to many different charities.
Think of it this way: To one extent or another, most of us care about large public goods such
as finding cures to diseases, alleviating poverty, saving the environment, etc. But, aside from the
Bill Gates’ of the world, most of us have modest resources to contribute to solving these very large
problems. If all we care about as we contemplate how much and to whom to give, the rational
course of action would be to find the public good that we care about most and where we think our
contribution can have the biggest impact. We should then give the entire amount that we decide to
devote to charitable purposes to one and only one cause. Suppose, for example, I care most about
poor children in the developing world and I want to make as much of a difference there as I can.
7 The “light dues” that funded lighthouses across England, Scotland and Whales were collected by customs officials
in ports – which created the effective bundling of port use to use of lighthouses. For a detailed discussion of this, see
Coase, R. (1974) “The Lighthouse in Economics,” The Journal of Law and Economics 17(2), 357-76.
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Chapter 27. Public Goods
Once I have given $1,000 or $10,000 to that effort, it is hard for me to think that I have now made
enough of a difference in alleviating poverty in the developing world to now move on to contribute
my next dollar to a different public good – say Alzheimer research or the local Girl Scouts. I am
simply too small a part of the world for my contribution to make a large enough marginal impact
in the area I care about most to think I have “solved” that problem sufficiently to move onto the
next one.
But in most cases, we actually see individuals giving their time and money to multiple causes.
A model of giving that assumes we only take into account the difference our giving makes in the
world cannot rationalize this behavior. So when I see others (or myself) giving to multiple causes,
there most be something else that explains this pattern of giving, just as there must be something
else that explains why we give as much as we do. And that “something else” often has to do with
the way that Civil Society institutions persuade us to give. In some instances, as we will see, we
might be seeing the Coase Theorem (that we introduced in Chapter 21) at work, and in other
cases civil society institutions persuade us that we in fact get private benefits in addition to the
public benefit from our giving. In this section, we’ll further explore these ways in which the civil
society engages – and why it sometimes succeeds so much more than other times. Finally, civil
society institutions might design creative incentive schemes that overcome the prisoners’ dilemma
incentives. In end-of-chapter exercise 27.5, we give an example of this in the context of a particular
type of fundraising campaign that some civil society institutions employ.
27A.4.1
Small Public Goods and The Coase Theorem
In Chapter 21, we introduced the Coase Theorem in the broader context of externalities, and
we illustrated Coase’s argument that, as long as property rights are sufficiently well defined and
transactions costs are sufficiently low, decentralized bargaining would result in optimal outcomes.
We developed the theorem for the case of negative externalities, but the same argument holds for
positive externalities (such as those produced by public goods).
Suppose we think again of you and me launching fireworks. In this case, the property rights are
pretty settled: You have the right to enjoy my fireworks without paying for them (and I have the
right to enjoy yours). If I take you to court to demand compensation for the enjoyment you get
from my fireworks, the court will probably give me a swift kick and tell me to go away. I therefore
have an incentive to go over to your house for coffee to discuss the whole fireworks issue and to
see if we can’t find a way for you to contribute so that we can jointly find a way out of our little
Prisoners’ Dilemma. If transactions costs – including the costs of enforcing our agreement – are
sufficiently low, we should be able to solve our dilemma.
This might help explain why we often voluntarily provide for multiple public goods in our
immediate vicinity, especially when we combine our understanding of the Coase Theorem with the
intuitions from our game theory chapter that suggest cooperation between players with Prisoners’
Dilemma incentives can emerge in settings where they interact repeatedly (and each time believe
there is a good chance they will meet again). But it cannot get us very far toward explaining why
we give to larger public goods the way we do – to museums, universities, hospitals and perhaps
even economics departments.
27A.4.2
Private Benefits from Public Giving: The “Warm Glow Effect”
Suppose that I write checks to support Alzheimer research not only because I believe that my check
will have a positive marginal impact on the probability that a cure will be found but also because,
27A. Public Goods and their Externalities
1077
whenever I write such a check, I remember my grandmother who passed away from this dreadful
disease and I take pleasure in remember (and honoring) her through my contribution. In such cases,
economists say that I am deriving a “warm glow” from giving to a public good – I feel good even
if my contribution actually does nothing to get us closer to a cure for Alzheimers. Put differently,
I get a private benefit from my public giving. And to the extent to which our purpose for giving to
charitable causes fulfills a private need, we do not encounter the free-rider problem any more than
we do when we think of my “contribution” to buying my lunch. While the free rider problem is
still present to the degree to which Alzheimer research is a public good, it is counteracted by the
private benefit I receive from writing my check. And the more the Alzheimer Research Foundation
can get me to view my contribution as honoring my grandmother rather than contributing to the
big public good of finding a cure, the smaller is the free rider problem that remains to be overcome.
In the case of my contributions to Alzheimer’s research, there are particular reasons for my
“warm glow”, but in other cases charitable organizations deliberately manufacture such reasons in
the way they market themselves. In a previous chapter, we mentioned the case of relief organizations
that help poor families and communities in developing countries. You have almost certainly seen
such agencies advertise that, with a monthly contribution of $20, you can change a particular
child’s life. Not only that, the organization will match you with a particular child and establish
contact with the family, send you pictures and yearly updates, etc. It seems highly unlikely that
such organizations will actually stop helping a particular family if you stop sending checks – which
means that your contribution is actually a contribution to a larger “public good” of alleviating
poverty in the developing world. But by framing their fundraising efforts in a way that personalizes
your contributions, the organizations in essence attempts to convert what is a fairly abstract public
good to a concrete private good – helping one particular family that you end up caring about. It is,
in the language we used in Chapter 26, an example of “image marketing” in which the organization
changes the image of what it is asking you to contribute to in order to make it more likely that you
will view your contribution as a private rather than a public good.
Exercise 27A.24 Explain how it is rational for me to give to both relieving poverty in the third world and
to Alzheimer research in the presence of “warm glow” but not in its absence.
Non-profit organizations can therefore make use of image marketing just as for-profit firms
do, except that we tend to think of successful image marketing that leads to greater charitable
giving as a socially positive outcome given that it helps individuals overcome Prisoners’ Dilemma
incentives. Churches appeal to a sense that we are working toward a reward in the next life as we
give “selflessly” in this life; local relief organizations offer individuals a chance to build meaningful
relationships as they volunteer to build houses for the homeless; universities put names of large
donors on buildings to give a private reward for giving to a public good; and public radio stations
give bumper stickers to contributors so that they can proudly display these on their cars. There
is nothing in any of these efforts to guarantee an “optimal” level of public goods provision within
the civil society, but all of them appear to succeed in overcoming Prisoner’s dilemma incentives to
some extent through providing contributors with a “warm glow” from giving.
From Coase we learn that it is important to have individuals “take ownership” of externalities,
and that it is similarly important to insure that transactions costs of people taking such ownership
are low. One way to think of civil society efforts to provide public goods is to then think of such
organizations as finding creative ways of getting individuals to “take ownership” and reducing the
transactions costs of participating in the lowering of externality inefficiencies when such ownership
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Chapter 27. Public Goods
has been established. Linking your contribution to the alleviation of poverty in the developing
world to a particular family you are supposedly helping is a way of establishing ownership in the
presence of a desire by individuals to “make a difference”. It is also a way of having an organization
take on the task of coordinating the efforts of many individuals and thereby reduce the transactions
costs individuals would face in the absence of such civil society institutions.
Exercise 27A.25 Can you use the “warm glow effect” to explain why government contributions to public
goods (such as public radio) do not fully crowd out private contributions?
27A.4.3
Civil Society, Warm Glows and “Tipping Points”
And then there are the occasional episodes in history when very large public goods appear to emerge
quite spontaneously from civil society interactions outside government or market mechanisms. We
can think, for example, of the big social movements of the past century – civil rights marches in the
1960s when white and black Americans gave up their time (often at considerable risk) to demand
social chance. Or one can think of the Solidarity movement in Poland that laid the foundation for
the fall of the Iron Curtain in Eastern Europe. Or the demonstration of ”people power” that drove
dictators in places like the Philipines into exile. Such large social movements often aim toward
social change that affects us all – and as such represent attempts to provide large public goods
(like more democracy, more human rights, etc.). But most of our models would suggest that such
movements are unlikely to gain much momentum – because the larger they get, the deeper the free
rider problem they encounter. Does it really make sense for me to skip work or a day in the park
with my family to go to a rally in which millions are already participating? Is there any chance
that my contribution to the rally will make any difference whatsoever?
And yet, under some circumstances, individuals seem to be willing to risk almost anything to
be a part of such movements – and on occasion, such movements have established public goods
(such as greater civil rights) quite successfully without (and often in spite of) government action
(or inaction). One theory that explains such phenomena is based on an assumption that we derive
increased private benefits from participating in such movements the more of our friends participate.
(We previously encountered this idea in some of our Chapter 21 end-of-chapter exercises where
we modeled such network externalities in business and policy settings.) Someone who feels really
strongly about a particular issue might start standing on a street corner, and most of the time
that’s pretty much where it ends. Maybe a few others who feel strongly about the issue (or who
just feel sorry for the guy) show some support and stand there with him. But sometimes, as others
join, yet others join and the movement builds into an avalanche that can’t be stopped. At a critical
point, such movements cross a “tipping point” where they gain a self-perpetuating momentum,
while movements that don’t cross the “tipping point” quickly fizzle and become remembered as
quaint fads.
Suppose individuals in some group (like a church congregation) differ in their demand for a
public good y (like helping the poor) – but all individuals receive a greater warm glow from giving
to the public good the more others gave in the previous period (where you can think of a period
as a day or a week or a month, depending on the application). Such models tend to have at least
two pure strategy Nash equilibria. In one equilibrium, few people contribute and, because so few
people contribute, most people do not get much of a “warm glow” from contributing. In a second
equilibrium, most people contribute and, because so many contribute, people get substantial “warm
glow” from contributing. Social entrepreneurs (like the young idealistic minister that takes over
a congregation) therefore often have the challenge of starting in a low contribution equilibrium
27A. Public Goods and their Externalities
1079
and finding ways of getting sufficiently many individuals energized to cross a tipping point that
takes them to the high contribution equilibrium. They must first find those who are most deeply
committed and then hope that such individuals have sufficient social contacts with others who care
less about the public good at hand but who care more as the number of other people engaged in
the movement increases.8
Exercise 27A.26 * Suppose my warm glow from demonstrating in the streets (for some worthy cause)
depends on how much you demonstrate in the streets and vice versa. Letting the fraction of our time spent
demonstrating go from 0 to 1, suppose that I do not get enough of a warm glow from demonstrating unless
you spend at least half your time on the streets, and you feel similarly (about your warm glow and my
participation). Illustrate our best response functions to each other’s time on the streets. Where are the two
stable pure strategy Nash equilibria – and where is the tipping point?
27A.5
Preference Revelation Mechanisms
The problem of providing public goods optimally could, as we saw at the beginning of the chapter,
be easily solved if we just knew people’s preferences for public goods. We would then simply have
to add up individual demands and find where the aggregate demand for public goods crosses the
marginal cost of providing such goods. We could then also implement Lindahl prices for public
goods – which would ensure that individuals are charged appropriately for the marginal benefits
they receive from the optimal level of public goods we provide. But, as we saw in our discussion of
Lindahl pricing, we face a fundamental underlying problem: Individuals typically have an incentive
to misrepresent their preferences for public goods if their contributions to the public good are linked
to their stated preferences for public goods. Economists have therefore thought hard about how to
overcome this problem, and they have proposed “mechanisms” that take into account this incentive
problem. The general study of creating mechanisms that provide individuals with the incentive to
truthfully reveal private information (like their preferences for public goods) is called mechanism
design.
The fundamental problem faced by mechanism designers is the following: The designer has a
clear idea of what he would like to do if he could magically know people’s preferences. But since
he does not know those preferences, he needs to come up with an incentive scheme that makes it in
people’s best interest to tell the mechanism designer their true preferences. And this scheme has to
be such that individuals think it is in their best interest to reveal information truthfully as they take
into account what the mechanism designer will do with the information he collects. In the public
goods context, the mechanism designer would like to know people’s preferences over public goods
in order to implement the optimal public goods level. So what he needs to do is define “messages”
that individuals can send him and that contain the information he needs to determine optimal
public good provision, and then he needs to define a method by which he uses these “messages” to
determine how much public good to produce. That “method” in turn needs to have the property
that it provides individuals the incentive to send true messages about their tastes for public goods.
Exercise 27A.27 Suppose you have a piece of art that you would like to give to the person who values it
the most but you do not know people’s tastes. Explain how a second-price sealed bid auction (as described
8 This theory of multiple equilibria and tipping points applies not only to social movements and related contributions to public goods. For a fascinating discussion of how tipping points between low and high equilibria emerge
in all sorts of interesting circumstances, I highly recommend reading the recent best-seller Gladwell, Malcom (2000)
Tipping Point How Little Things Can Make Big Difference, Little Brown and Company: New York, NY.
1080
Chapter 27. Public Goods
in exercise 24.10) represents a mechanism that accomplishes this while eliciting truthful messages from all
interested parties.
27A.5.1
A Simple Example of a Mechanism
Suppose that you and I live at the end of a culdesac that currently has no street light. At night it
gets very dark in front of our houses and we therefore approach the city government about putting
up a light. The city would like to help but only if the value that you and I place on the street light
actually exceeds the cost of $1,000 it will cost to put it up. We do our best to use artful prose
to verbalize our deep desire to have light, punctuated by an occasional reference to our phobias of
darkness. But the city knows we have every incentive to exaggerate our desire for light and fear of
the dark in order to get the taxpayers to fund the light on our street. The city therefore needs to
figure out a way for us to reveal our true desires.
So, the mayor proposes the following. To begin with, he splits the $1,000 cost in two and asks
us each to write him a check for $500. He then asks us to tell him how much value above (or below)
$500 we each place on the street light. In other words, he asks us to send him a “message” that is
simply a number – which could be negative (if we want to tell him we place less that $500 value
on the light) or positive (if we want to tell him we place more than $500 of value on the light).
Let’s denote the message that you and I send as m1 and m2 respectively. The city will only build
the light if we indicate the value we place on the light is at least $1,000. Since the messages we
send are messages about how much each of us values the light above $500, this means the city will
only build the street light if m1 + m2 ≥ 0. The mayor furthermore tells us that, if the city ends up
building the street light, he will refund me an amount equal to the message m1 that you sent while
refunding you an amount equal to the message m2 that I sent. If you send a message m1 > 0, I
will therefore get a partial refund, but if you send a message m1 < 0, I will have to write another
check for the amount (−m1 ). If, on the other hand, the city does not build the street light (because
m1 + m2 < 0), the mayor will refund our $500 checks.
The city has therefore set up a simultaneous move “message sending” game in which each of us
now has to decide what message to send about our true underlying preference for the street light.
Let v1 and v2 denote your and my true valuation of the light above $500. If the light is built, you
will therefore get your true value v1 from enjoying the street light beyond the $500 payment you
have made plus you will get a check from the mayor equal to m2 if m2 > 0 or you will have to
write another check equal to (−m2 ) if m2 < 0. Your total “payoff” if the street light is built is
therefore (v1 + m2 ), while your total “payoff” if the street light is not built is 0 (since your $500
will be refunded).
At the time you decide what m1 message to send to the mayor, you do not know what m2
message I am sending. It may be that −m2 ≤ v1 or it may be that −m2 > v1 . If −m2 ≤ v1 , we can
add m2 to both sides of the inequality and get v1 + m2 ≥ 0. Thus, if you send a truthful message of
m1 = v1 , m1 + m2 ≥ 0 and the street light will be built. Your resulting payoff is then v1 + m2 ≥ 0
which is at least as good as getting a payoff of 0 that would occur if you sent a false message that
caused the light not to be built. Thus, if −m2 ≤ v1 , you should send a truthful message m1 = v1 .
Now suppose the other scenario is true – i.e. −m2 > v1 . If, under that scenario, you again sent
a truthful message m1 = v1 , then m1 + m2 < 0, the street light does not get built and you get a
payoff of 0. If you instead sent a false message that is high enough to get the street light built,
your payoff will be v1 + m2 < 0 – so again it’s best to send the truthful message m1 = v1 . Thus,
regardless of what message m2 I send, it is your best strategy to send a truthful message about your
27A. Public Goods and their Externalities
1081
own preferences. Put differently, truth telling in this game is a dominant strategy. Since I face the
same incentives as you, we will both send truthful messages and the street light gets built only if
we value the light more than what it costs.
If there are N > 2 people at the end of the culdesac, the city can design analogous mechanisms
that will similarly result in truth-telling. Instead of beginning with a charge of $500 for each person,
the city would instead charge each person $1,000/N at the beginning and build the light only if the
sum of the messages is at least zero. It would then refund to each person an amount equal to the
sum of the other people’s messages.
Exercise 27A.28 Suppose three people lived at the end of the culdesac and suppose the mayor proposes the
same mechanism except that he now asks you for a $333.33 check at the start (instead of $500) and you are
told (as player 1) that you will get a refund equal to m2 + m3 if (m1 + m2 + m3 ) ≥ 0 and the light is built.
(Otherwise, you just get your $333.33 back and no light is built.) Can you show that truth telling is again
a dominant strategy for you?
27A.5.2
Truth-telling Mechanisms and their Problems
We have therefore given a simple example of a mechanism in which the government elicits the
necessary information to determine whether a public good should be built. The trick for doing this
was that the payoff to each of the people does not depend on the message they send except to the
extent that each person’s message might be pivotal in determining whether or not the public good
is provided. Remember – your payoff was constructed to be equal to v1 + m2 if the street light is
built and 0 otherwise. Nowhere in your payoffs does your own message m1 appear – it only matters
in the sense that it enters the city’s decision on whether or not to put up the street light. So all you
had to think about was whether it made sense to tell the truth knowing that this will determine
whether the street light is built, and in making that decision the city forced you to consider the
messages sent by others about how much they value the street light. Put differently, the mechanism
we designed forces you to consider in your own decision how much others value the street light – by
making a payment to you that equaled the sum of how much (above $500) other people said they
valued the light.
Of course, the typical public goods decision is not whether to provide a public good but also
how much of the public good to provide. A city, for instance, has to decide how much police to
hire to insure public safety, and a higher level government has to decide how much to spend on
national defense. In Section B, we will illustrate a different version of the simple mechanism we just
discussed, a version that will permit the determination of the optimal quantity of a more continuous
public good, and again we will find a way to get people to tell the truth about their preferences.9
A second problem with our simple mechanism is that it will generally not yield sufficient revenues to fund the public good. Thus, while the mechanism elicits truthful information for the city
to determine whether or not to invest in the public good, it does not provide sufficient funds for
actually paying the cost. This, too, is a problem that is addressed in the somewhat more elaborate
mechanism introduced in Section B where we will present a mechanism that elicits truthful information and generates at least as much revenue as will be necessary to fund the optimal public good
level.
9 Our discussion of the more elaborate mechanism in Section B is relatively non-mathematically and can be
understood solely based on the graphs in that section. The interested reader can therefore investigate this mechanism
further without the mathematical background that is generally presumed for B-portions of our chapters.
1082
Chapter 27. Public Goods
Exercise 27A.29 Can you think of a case where our simple mechanism generates sufficient revenues to
pay for the street light?
Exercise 27A.30 Can you think of a case where the mechanism results in an outcome under which the city
needs to come up with more money than the cost of the street light in order to implement the mechanism?
More generally, as we further discuss in Section B, preference revelation mechanisms cannot
implement (and fund) fully efficient outcomes if our goal is to have truth telling be a dominant
strategy (Nash) equilibrium, but they can do so if we only require truth telling to be a Nash
equilibrium strategy. For now, the main point to take away from our discussion is that we can
think of mechanisms to elicit truthful information about public goods preferences and thereby
overcome the incentive to misrepresent preferences in order to free-ride on others. However, such
mechanisms come at a cost that might make it difficult to implement them in many circumstances.
In fact, such mechanisms have only been used on rare occasions to provide public goods.
27A.5.3
Mechanism Design More Generally
Not all mechanisms, however, have as their goal to provide public goods. There are, as we have
seen before in this book, many circumstances where some parties have more relevant information
than others that would like to acquire some of that information. In such cases, mechanisms can
be designed to get individuals to reveal private information knowing what will happen once that
information is revealed. Economists, for instance, have had major roles in designing mechanisms
by which large public holdings are auctioned in ways that reveal the private valuations by bidders
for the public holdings. Economists have also designed mechanisms that, in the absence of market
prices, result in optimal “matches” between buyers and sellers. For instance, the mechanism that
determines which hospitals are matched with which medical school interns is one that has been
designed by economists, as have new mechanisms to match live kidney donors with patients. (The
problem in kidney donations is that I might be willing to donate a kidney to my relative and you
might be willing to donate your kidney to your relative but neither one of us has the right kidney
for the person to whom we are trying to donate. If my kidney is a good match for your relative
and yours is a good match for mine, however, there is still a way for our relatives to get donated
kidneys if we can find the right mechanisms to determine how such matches are to be made.) In
the past few years, economists have also designed large public school choice programs in cities like
Boston and New York – programs where parents provide information about their preferences for
schools and the mechanism then matches children to schools.10 While it is beyond the scope of this
text, the general area of mechanism design is therefor one of growing interest among economists
who aim to achieve more efficient outcomes in the real world when markets on their own cannot
get there. It is a fascinating area that you might want to study more.
27B
The Mathematics of Public Goods
We begin our mathematical treatment of public goods in Section 27B.1 by illustrating the basic
necessary condition for public good quantities to be optimal. While we do this for a general case
with many consumers, we then introduce a simple example involving two consumers with well
10 Much of this literature – and efforts to bring its results into the real world – are due to Alvin Roth (1951-),
an economics professor at Harvard, and a number of his notable collaborators. Interested students might consider
exploring some of Professor Roth’s website that overviews many recent developments.
27B. The Mathematics of Public Goods
1083
defined and identical preferences, and we will use this example throughout the chapter to illustrate
the mathematics behind the intuitions developed in Section A. As in our intuitive development
of the material, we will demonstrate the free rider problem as an outgrowth of the presence of
positive externalities that individuals generally do not take into account unless their choices are
tempered by non-market institutions. The direct government policies of public good provision
and public good subsidies are introduced in Section 27B.2, and the more indirect “policies” of
establishing certain types of markets are discussed in Section 27B.3. Section 27B.4 then considers
civil society intervention, particularly in the presence of “warm glow” effects of giving, and Section
27B.5 expands our discussion of preference revelation mechanisms from the simple mechanism
discussed in Section A.
27B.1
Public Goods and the Free Rider Problem
Public goods, as we have seen, give rise to externalities, and we already know from earlier chapters
that decentralized market behavior in the presence of externalities often does not result in efficient
outcomes. We begin by deriving the necessary condition for optimality of public goods – the
condition now quite familiar (from our work in Section A) that the sum of marginal benefits must
equal the marginal cost of producing the public good. We then proceed, as we did in Section A, to
illustrate the free rider problem that keeps decentralized market behavior from being efficient.
27B.1.1
The Efficient Level of Public Goods
Suppose x represents a composite private good and y represents the public good. There are N
consumers in the economy, with un (xn , y) representing the nth consumer’s preferences over her
consumption of the composite private good xn and the public good. Suppose further that f represents the technology for producing y from the composite good; i.e. suppose y = f (x). Finally,
suppose that the total available level of private good (in the absence of public goods production)
is X.
We are first interested in deriving the necessary conditions that have to be satisfied for us to
produce an efficient public good level y ∗ . For a situation to be efficient, we have to set y ∗ such
that nothing can be changed to make one consumer better off without making some consumers
worse off. We can therefore calculate this by choosing the consumption levels (x1 , x2 , ..., xN ) and y
to maximize one consumer’s utility subject
Pto holding the others fixed at some arbitrary level and
subject to the constraint that y = f (X − xn ).
Exercise 27B.1 Explain the constraint y = f (X −
P
xn ).
To cut down a bit
P on notation as we write
P down this optimization problem formally, we can
define a function g( xn , y) = y − f (X − xn ). We can then formally express the optimization
problem to derive the necessary conditions for an efficient public good level y ∗ as
max
(x1 ,...xN ,y)
u1 (x1 , y) subject to un (xn , y) = un for all n = 2, ..., N and g
The Lagrange function for this optimization problem is
N
X
n=1
xn , y
!
= 0. (27.1)
1084
Chapter 27. Public Goods
L = u1 (x1 , y) +
N
X
λn (un − un (xn , y)) + λ1 g
N
X
xn , y ,
n=1
n=2
!
(27.2)
where (λ2 , ...λN ) are the Lagrange multipliers for the constraints that hold utility levels for
consumers 2 through N fixed and λ1 is the Lagrange multiplier for the production constraint. To
get our first order conditions, we differentiate L with respect to each of the choice variables to get
∂g
∂u1
∂L
=
+ λ1
=0
∂x1
∂x1
∂x
∂L
∂un
∂g
= −λn
+ λ1
= 0 for all n = 2, ..., N
∂xn
∂xn
∂x
N
∂g
∂u1 X ∂un
∂L
=
−
λn
+ λ1
= 0,
∂y
∂y
∂y
∂y
n=2
(27.3)
where we can express ∂g/∂xi simply as ∂g/∂x (since marginal increases in any xi have the same
impact on the first argument of the g function). The first of our first order conditions can be written
as ∂u1 /∂x1 = −λ1 ∂g/∂x. We can then divide the first term of the third first order condition by
∂u1 /∂x1 and the remaining terms by −λ1 ∂g/∂x. Subtracting the resulting last term from both
sides, the last first order condition becomes
N
X λn ∂un /∂y
∂u1 /∂y
∂g/∂y
+
=
.
∂u1 /∂x1 n=2 λ1 ∂g/∂x
∂g/∂x
(27.4)
λn
∂g/∂x
=
for all n = 2, ..., N,
λ1
∂un /∂xn
(27.5)
The second set of first order conditions can be re-written as
which, when substituted for λn /λ1 in equation (27.4), yields
N
X ∂un /∂y
∂g/∂y
∂u1 /∂y
+
=
.
1
∂u /∂x1 n=2 ∂un /∂xn
∂g/∂x
(27.6)
The first term in this equation can then be brought into the summation in the second term, and
the resulting equation can be inverted and multiplied by -1 to yield
N
X
n=1
−
∂un/∂xn
∂g/∂x
=−
.
∂un /∂y
∂g/∂y
(27.7)
Now notice that the left hand side of the equation is simply the sum of the marginal rates of
substitution for all the consumers in the economy – or the sum of the marginal benefits expressed
in dollars since we are interpreting x as a dollar-denominated composite
good. The right
P
P hand side
of the equation can be simplified given that g was defined as g( xn , y) = y − f (X − xn ), with
∂g/∂y = 1 and ∂g/∂x = ∂f /∂x. The right hand side therefore simplifies to ∂f /∂x, which is just
the marginal cost (in terms of x) of producing one more unit of y. Equation (27.7) can then simply
be written as
27B. The Mathematics of Public Goods
1085
N
X
M Byn = M Cy ,
(27.8)
n=1
i.e. the sum of the marginal benefits of the public good must be equal to the marginal cost of
producing it.11
27B.1.2
A Simple Example
To make this more concrete in the context of an example we will continue to use in other parts of
this section, suppose that we have an economy of 2 consumers who have identical Cobb-Douglas
preferences that can be represented by the utility function
1−α
un (xn , y) = xα
.
ny
(27.9)
Suppose further a simple production technology y = f (x) = x that permits us to produce 1 unit
of the public good from 1 unit of the composite private good, and suppose the only resources we
have are the incomes of the two consumers, I1 and I2 .
To find the efficient level of the public good y ∗ , we can then again calculate this by choosing x1 ,
x2 and y to maximize one consumer’s utility subject to holding the other’s fixed at some arbitrary
indifference curve u and subject to the constraint that only the consumers’ incomes can be used to
fund the public good; i.e we can solve the optimization problem
max u1 (x1 , y) subject to u2 (x2 , y) = u and y = (I1 + I2 − x1 − x2 ).
x1 ,x2 ,y
(27.10)
It is easiest to solve this by taking natural logarithms of the utility function and substituting
y = (I1 + I2 − x1 − x2 ) into the utility functions for y. We can then write the optimization problem
as
max α ln x1 +(1−α) ln(I1 +I2 −x1 −x2 ) subject to α ln x2 +(1−α) ln(I1 +I2 −x1 −x2 ) = u. (27.11)
x1 ,x2
Solving the two first order conditions, we get
x1 + x2 = α(I1 + I2 )
(27.12)
y ∗ = I1 + I2 − x1 − x2 = (I1 + I2 ) − α(I1 + I2 ) = (1 − α)(I1 + I2 ).
(27.13)
which implies
Exercise 27B.2 Verify the outcome of this optimization problem. (Hint: Solve the first two first order
conditions for λ and use your answer to derive the equation for (x1 + x2 ).)
11 The optimality condition for public goods is often referred to as the “Samuelsonian” optimality conditions because
of their original formal derivation by Paul Samuelson (1915-), the 1970 winner of the Nobel Prize in Economics.
Samuelson, an economics Professor at MIT, was only the second economist to be awarded a Nobel Prize following
the creation of the Prize in 1969.
1086
Chapter 27. Public Goods
We can also check that this is the optimal quantity of public goods by adding up demand curves
as we did in Section A. We know that Cobb-Douglas preferences represented by u(x, y) = xα y (1−α)
give rise to demand curves for y of the form y = (1 − α)I/p. Writing this as an inverse demand
curve, consumer n’s demand is p = (1 − α)In /y. If we consider two consumers with identical
preferences but different incomes, the (vertical) sum of these is
(1 − α)I2
(1 − α)(I1 + I2 )
(1 − α)I1
+
=
.
y
y
y
(27.14)
When the production technology for y takes the simple form y = f (x) = x, the marginal cost
of producing 1 additional unit of y is c = 1. Thus, a social planner who is interested in providing
the efficient level of the public good would produce y so long as equation (27.14) is greater than
marginal cost and would stop when
(1 − α)(I1 + I2 )
= 1.
y
(27.15)
Solving for y, we again get the optimal level of public goods as
y ∗ = (1 − α)(I1 + I2 ).
(27.16)
Exercise 27B.3 What is y ∗ if there are N rather than 2 consumers of the type described in our example
(i.e. with the same Cobb-Douglas tastes but different incomes)? What if everyone’s income is also the
same?
27B.1.3
Decentralized Provision of Public Goods
Suppose we now continue with our example and we ask the two consumers to voluntarily contribute
to the provision of the public good. In other words, suppose we asked each consumer n to decide
on a contribution zn of her income (or the composite good), with each consumer knowing that the
public good y will be a function of their joint contributions such that
y(z1 , z2 ) = z1 + z2 .
(27.17)
The consumers are then engaged in a simultaneous move game in which they both choose their
individual contributions taking the other’s contribution as given. To determine consumer 1’s best
response function to consumer 2 contributing z2 , consumer 1 would solve the problem
max u1 (x1 , y) such that I1 = x1 + p1 z1 and y = z1 + z2 ,
x1 ,z1
(27.18)
where we have implicitly assumed that the price of x is 1 since x is a dollar-denominated
composite good. We have also assumed a “price” pn for contributing to the public good, where
pn is equal to 1 if no one is subsidizing the contributions of individuals. (We are including the
possibility of subsidies in preparation for discussing government subsidies of private giving.)
Exercise 27B.4 Explain why p1 = p2 = 1 for both consumers in the absence of subsidies for giving to the
public good.
27B. The Mathematics of Public Goods
1087
Substituting y = (z1 + z2 ) for y and x1 = I1 − p1 z1 for x1 into the logarithmic transformation
of the Cobb-Douglas utility function from equation (27.9), the problem then becomes
max α ln(I1 − p1 z1 ) + (1 − α) ln(z1 + z2 ),
z1
(27.19)
where the first order condition now just involves taking the derivative of the utility function
with respect to z1 . Solving this first order condition then gives consumer 1’s best response function
to z2 as
z1 (z2 ) =
(1 − α)I1
− αz2 ,
p1
(27.20)
and doing the same for consumer 2 we can similarly get consumer 2’s best response function to
z1 as
z2 (z1 ) =
(1 − α)I2
− αz1 .
p2
(27.21)
Exercise 27B.5 Draw the best response functions for the two individuals in a graph similar to Graph 27.3.
Carefully label intercepts and slopes.
In a Nash equilibrium to this game, each consumer has to be best responding to the other.
Plugging equation (27.21) in for z2 in equation (27.20), we can solve for consumer 1’s equilibrium
contribution as
z1eq =
I1 p2 − αI2 p1
(1 + α)p1 p2
(27.22)
and plugging this back into equation (27.21), we get consumer 2’s equilibrium contribution as
z2eq =
I2 p1 − αI1 p2
.
(1 + α)p1 p2
(27.23)
The sum of the individual contributions, and thus the equilibrium level of the public good under
voluntary giving y v , is therefore
y v (p1 , p2 ) = z1eq + z1eq =
(1 − α)(I1 p2 + I2 p1 )
.
(1 + α)p1 p2
(27.24)
Now suppose that consumers in fact do not receive any subsidy to give to the public good, which
implies p1 = p2 = 1. Then equation (27.24) simplifies to
y v (no subsidy) =
(1 − α)(I1 + I2 )
< (1 − α)(I1 + I2 ) = y ∗
(1 + α)
(27.25)
where the inequality holds for all α > 0. Thus, so long as consumers place at least some value
on private good consumption, the voluntary contributions result in less than the optimal quantity
of the public good as each consumer free-rides on the contributions of the other.
Exercise 27B.6 Why do private contributions to the public good result in the optimal level of the public
good when α = 0?
1088
Chapter 27. Public Goods
y eq
y∗
eq
y /y ∗
Free Riding as Population Increases
N =1
N =2
N =5
N =10 N =25
500
666.67 833.33 909.09 961.54
500
1,000
2,500
5,000 12,500
1.000 0.667
0.333
0.182 0.077
N =100
990.10
50,000
0.020
Table 27.2: I = 1, 000, α = 0.5
Exercise 27B.7 Consider the equilibrium public good level as a fraction of the optimal public good level.
In our example, what is the lowest this fraction can become, and what is the critical variable?
You can easily see how this under-provision of public goods under voluntary giving will continue
(and in fact get worse) as the number of consumers increases. Suppose, for instance, that everyone
is identical in every way – both in terms of their Cobb-Douglas preferences and in terms of their
income, and that there is no subsidy for private giving to charity. But now instead of two of us
there are N of us. In a symmetric equilibrium (in which all the identical players play the same
strategy), we can then simplify equation (27.20) to
z = (1 − α)I − α(N − 1)z,
(27.26)
where (N − 1)z is the contribution by all (N − 1) players other than the one whose best response
function we are working with. Solving this for z, we get
z eq =
(1 − α)I
,
1 + α(N − 1)
(27.27)
and the resulting equilibrium level of public good y eq is simply equal to N z eq or
y eq =
N (1 − α)I
.
1 + α(N − 1)
(27.28)
In exercise 27B.3, you should have derived the optimal level of the public good for the N -person
case as y ∗ = N (1 − α)I, which means we can re-write equation (27.28) as
y eq =
y∗
.
1 + α(N − 1)
(27.29)
An increase in the number of consumers N of the public good increases the denominator of the
right hand side of this equation, which means that as N increases, the equilibrium quantity of the
public good will be a decreasing fraction of the optimal quantity. Put differently, the free rider
problem gets worse as the number of consumers of the public good increases.
Table 27.2 demonstrates this dramatically for the case were all consumers have income I = 1, 000
and α = 0.5. The last row of the table reports the equilibrium public good level as a fraction of the
optimal public good level. This is 1 when there is only a single consumer (in the first column) and
there thus does not exist a free rider problem. But if falls quickly as we add consumers, already
reaching 0.02 at N = 100.
Exercise 27B.8 As N gets larger, what do y ∗ and y eq converge to for the example in Table 27.1? What
does the equilibrium level of public good as a fraction of the optimal level converge to?
27B. The Mathematics of Public Goods
27B.2
1089
Direct Government Policies to address Free Riding
As in Section A, we’ll consider two direct approaches a government might take to the public goods
problem. First, it may itself provide the public good, and second it may use subsidies to make it
cheaper for individuals to give to public goods. To result in optimal levels of the public good, both
approaches require knowledge of consumer preferences (which governments typically do not have,
a topic we take up again in Section 27B.5).
27B.2.1
Government Provision and “Crowd-out”
We have already seen how an efficiency-focused government would calculate the optimal level of
public goods, and in end-of-chapter exercise 27.9 you can show how this is affected if we also consider
the government needs to raise the necessary revenues to fund public good production when it can
only use inefficient taxes. Now suppose the government, either because it does not have sufficient
information about preferences or because the political process is not efficient, decides to fund some
amount g of the public good (rather than the optimal qantity y ∗ ), and suppose it funds this through
a proportional income tax t. Since income is assumed to be exogenous (and not the result of an
explicit labor-leisure choice), such a tax would have no dead weight loss in our example. In order
to raise sufficient revenues to fund g, it must be that t(I1 + I2 ) = g or, rearranging terms,
t=
g
.
(I1 + I2 )
(27.30)
Exercise 27B.9 Can you explain in a bit more detail why the tax in this case is efficient?
Each consumer n then has to determine how much zn to give to the public good herself given that
the government is contributing g. Consumer 1 therefore takes as given consumer 2’s contribution
z2 as well as the government contribution g, which changes the optimization problem in equation
(27.19) to
max α ln((1 − t)I1 − p1 z1 ) + (1 − α) ln(z1 + z2 + g),
z1
(27.31)
or, substituting in for t,
max α ln
z1
(I1 + I2 − g)I1
− p1 z1 + (1 − α) ln(z1 + z2 + g).
I1 + I2
(27.32)
Solving the first order condition for z1 , we get consumer 1’s best response to (z2 , g) as
z1 (z2 , g) =
(1 − α)I1 (I1 + I2 − g)
− α(z2 + g).
(I1 + I2 )p1
(27.33)
Similarly, consumer 2’s best response to (z1 , g) is
z2 (z1 , g) =
(1 − α)I2 (I1 + I2 − g)
− α(z1 + g).
(I1 + I2 )p2
(27.34)
Exercise 27B.10 Demonstrate that these best response functions converge to those in equations (27.20)
and (27.21) as g goes to zero.
1090
Chapter 27. Public Goods
Substituting consumer 2’s best response function into consumer 1’s and solving for z1 , we get
consumer 1’s equilibrium contribution to the public good as a function of the government’s contribution
αg
(I1 + I2 − g)(I1 p2 − αI2 p1 )
−
,
(1 + α)(I1 + I2 )p1 p2
(1 + α)
with consumer 2’s equilibrium contribution coming to
z1eq (g) =
(27.35)
(I1 + I2 − g)(I2 p1 − αI1 p2 )
αg
−
.
(27.36)
(1 + α)(I1 + I2 )p1 p2
(1 + α)
Adding these individual contributions to the government’s, we get the equilibrium public good
level y eq (g) as
z2eq (g) =
y eq (g) = z1eq (g) + z2eq (g) + g
2α
(1 − α)(I1 p2 + I2 p1 )
(1 − α)(I1 p2 + I2 p1 )
+g
−g
+
=
(1 + α)p1 p2
(1 + α)(I1 + I2 )p1 p2
1+α
(1 − α)(I1 p2 + I2 p1 )
2α
= yv + g − g
+
(1 + α)(I1 + I2 )p1 p2
1+α
(27.37)
where y v is our previous voluntary contribution level in the absence of government contributions (from equation (27.29)). When the government contributes $1 to the public good, private
contributions therefore decline by an amount equal to the bracketed term in the equation. Government contributions to the public good then crowd out private contributions dollar for dollar if
the bracketed term is equal to 1 which, you can check for yourself, occurs when p1 = p2 = 1. Put
differently, when the government is not subsidizing private contributions to the public good (and a
$1 in contributions costs $1), government contributions to the public good fully crowd out private
contributions.
Our perfect crowd-out result holds, however, only to the extent to which consumers are in fact
giving to the public good when the government increases its contribution. If a consumer is at a
“corner solution” where she does not give, then she remains at that corner solution as government
contributions rise. Consider, for instance, the simple case where the two consumers have identical
incomes I and where the government is not subsidizing individual contributions (i.e. p1 = p2 = 1).
Then equations (27.35) and (27.36) simply become
g
(1 − α)I
− .
1+α
2
This implies that individual contributions are zero when
z eq (g) =
(27.38)
2(1 − α)I
,
(27.39)
1+α
and for government contributions larger than this, there is no crowd-out. (In end-of-chapter
exercise 27.8 you can demonstrate that the same crowd-out result holds when the number of individuals is N instead of 2.)
g=
Exercise 27B.11 Can you tell if there is any crowd-out for the last dollar spent by the government if the
government provides the optimal level of the public good in this case?
27B. The Mathematics of Public Goods
27B.2.2
1091
Tax and Subsidy Policies to Encourage Voluntary Giving
Finally, suppose that the government wanted to offer a subsidy s to reduce the effective price that
individuals have to pay in order to contribute to the public good. They may do so directly or, as we
discussed in Section A, by making charitable contributions tax deductible. In order to finance this
subsidy, the government imposes a tax t on income, and since income is assumed to be exogenous,
such a tax would be efficient. By choosing a policy (t, s), the government therefore reduces consumer
n’s income to (1 − t)In and her price for contributing to the public good to (1 − s). Substituting
these new prices and incomes under policy (t, s) into equation (27.24), we can then write the total
amount of giving to the public good as
y v (t, s) =
(1 − α)(1 − t)(I1 + I2 )
(1 − α)[(1 − t)I1 (1 − s) + (1 − t)I2 (1 − s)]
=
.
(1 + α)(1 − s)2
(1 + α)(1 − s)
(27.40)
But the government can’t just pick any combination of t and s because it’s budget has to
balance. Put differently, tax revenues have to be sufficient to pay the subsidy. If the government
wants to set subsidies to induce the efficient level of the public good y ∗ = (1 − α)(I1 + I2 ), it knows
it must raise revenues equal to sy ∗ = s(1 − α)(I1 + I2 ). Its revenues are t(I1 + I2 ) – which implies
that, for a subsidy s that achieves the optimum level of public good y ∗ , the government needs to
set t such that
t(I1 + I2 ) = s(1 − α)(I1 + I2 )
(27.41)
which simplifies to t = s(1 − α). Substituting this into equation (27.40), we can write the level of
giving as a function of s assuming the government in fact balances its budget and sets t = s(1 − α);
i.e.
(1 − α)(1 − s(1 − α))(I1 + I2 )
.
(27.42)
(1 + α)(1 − s)
To insure the optimal level of contributions to the public good, it must then be that y v (s) = y ∗ ,
y v (s) =
or
(1 − α)(1 − s(1 − α))(I1 + I2 )
= (1 − α)(I1 + I2 ).
(27.43)
(1 + α)(1 − s)
With a little algebra, this solves to s = 1/2. Thus, the optimal combination of an income tax
and a subsidy for giving to the public good is
1−α 1
∗ ∗
.
(27.44)
(t , s ) =
,
2
2
In exercise 27.1 you can demonstrate that, in the N person case, the optimal subsidy level
becomes s∗ = (N − 1)/N (and in exercise 27.2 you can explore how the result changes if individuals
think more strategically about the balanced budget tax implications of their giving.)
Exercise 27B.12 Can you offer an intuitive explanation for why s∗ = 1/2? How would you expect this to
change as the number of consumers increases?
Exercise 27B.13 We previously concluded that the optimal level of the public good is (1 − α)(I1 + I2 ). Can
you use our solutions for s∗ and t∗ to show that this level is achieved through the voluntary contributions
of the 2 individuals when the policy (s∗ , t∗ ) is implemented?
1092
27B.3
Chapter 27. Public Goods
Establishing Markets for Public Goods
If we knew individual demands for public goods, we have seen that it would be easy to derive the
optimal public good quantity – and, as we saw in Section A, it would also be easy to then derive
personalized prices for different consumers, prices under which they would in fact choose the optimal
public good level that is simultaneously chosen by others (at their personalized prices) as well. This
notion of an equilibrium, called a Lindahl Equilibrium, is the public good analog to a competitive
private good equilibrium. It is, in some sense, the mirror image of our notion of a competitive
equilibrium where everyone faces the same prices and chooses different quantities – because in a
Lindahl equilibrium, everyone chooses the same quantities at different prices. We will begin below
by illustrating the mathematics of deriving the Lindahl equilibrium within our 2-person example
and then briefly move onto the case of local public goods.
27B.3.1
Lindahl Pricing and Markets for Public Good Externalities
Suppose a firm is producing the public good and selling it to consumer n at pn . The problem is
that the firm can only produce a single quantity of y that will be consumed by all consumers – and
so it looks for individualized prices such that (1) all consumers would in fact choose to purchase
the quantity y that is produced at their individualized price and (2) the producer covers her costs.
In order for the result to be efficient, it must further be the case that the quantity produced (and
demanded by each consumer) is y ∗ .
Given the simple production function y = f (x) = x, the producer faces a constant marginal
cost c = 1 for each unit of y she produces. Thus, to satisfy the condition that the producer’s costs
are covered (in the absence of fixed costs), it simply has to be the case that
p1 + p2 = 1.
(27.45)
We know from our work with Cobb-Douglas preferences that consumers will allocate a fraction
of their income to each consumption good, with that fraction being equal to the exponent that
accompanies that good in the utility function. Thus, we know that demand for y by consumer n is
yn =
(1 − α)In
.
pn
(27.46)
The price p∗n that will induce consumer n to purchase the optimal public good quantity y ∗ =
(1 − α)(I1 + I2 ) can therefore be determined by simply solving
(1 − α)(I1 + I2 ) =
(1 − α)In
pn
(27.47)
for pn . This gives us
p∗n =
In
.
I1 + I2
(27.48)
With each consumer being charged this price, the sum of the prices is 1 (thus satisfying condition
(27.45)) and each consumer chooses y ∗ = (1 − α)(I1 + I2 ).
Exercise 27B.14 What do you think pn will be in the N -person case if everyone shares the same CobbDouglas tastes? What if they also all have the same income level?
27B. The Mathematics of Public Goods
27B.3.2
1093
Local Public and Club Goods
An alternative “market” solution to (local) public goods provision involves, as we discussed in
Section A, having clubs or local communities compete for customers or residents when public goods
are excludable. Under conditions we explore further in end-of-chapter exercise 27.4, this results in
competition that is analogous to our notion of a competitive equilibrium, with individuals choosing
clubs and communities much as they choose supermarkets and shopping centers. The “Tiebout”
literature that explores these intuitions is vast, and a detailed mathematical exploration of the
properties of Tiebout models is beyond the scope of this text. The interested student should
consider taking courses in local public finance and urban economics.
27B.4
Civil Society and the Free Rider Problem
We noted in Section A that, if all we care about is the overall level of the public good but not how
that level was arrived at, we should almost never be observed to contribute to more than a single
charity. The intuition for this is straightforward: Our contributions to charities are almost always
small relative to the size of the public good that is being funded. This means that the marginal
impact of our contribution is unlikely to cause a sufficiently large change in the overall public good
to warrant switching charities. If charity A was the best charity to give to before I wrote my check,
it is still the best charity to give to after I write my check – because my check is simply not very
big compared to the overall need.
It is not difficult to see this mathematically. Suppose there are three charities – a, b and c,
and before I write my check, they have already received total contributions of Ya , Yb and Yc . As
I consider where to place my contribution, I have come to some judgment about how much these
charities add in value to the world, and I can represent this judgment by a function F (Ya , Yb , Yc ).
If I have an amount D to donate, I will then want to donate in a way that maximizes the impact I
have on the world based on my judgement F , i.e. I would like to solve the problem
max F (Ya + ya , Yb + yb , Yc + yc ) subject to D = ya + yb + yc
ya ,yb ,yc
(27.49)
where yi is my contribution to charity i. When D is small relative to each Yi , the only way that
I will arrive at an “interior solution” where yi > 0 for i = a, b, c is if, prior to my contributions,
∂F
∂F
∂F
=
=
.
(27.50)
∂Ya
∂Yb
∂Yc
In that case, I need to make sure that I “balance” my contributions so that this equation
continues to hold after I have contributed. But if ∂F/∂Ya is greater than ∂F/∂Yb and ∂F/∂Yc ,
then I will solve my optimization problem (27.49) by setting ya = D and yb = yc = 0 since it is
unlikely that my (relatively) small contribution lowers ∂F/∂Ya in any perceptible way. Notice that,
to the extent to which I am uncertain about the marginal impact my contributions will have across
charities, this is part of the F function that captures my judgments about where my contributions
will have their largest impact – and so uncertainty does not undo the argument that people should
give only to a single charity if they care only about the impact their contribution has on the world.
Exercise 27B.15 What is different for Bill Gates that might make him rationally contribute to multiple
charities?
Exercise 27B.16 Suppose I only give to small local charities. In what way might I then be like Bill Gates
and give rationally to more than one?
1094
Chapter 27. Public Goods
Exercise 27B.17 Can you explain why it is rational to diversify a private investment portfolio in the
presence of risk and uncertainty but the same argument does not hold for diversifying our charitable giving?
Given how often we see individuals give relatively small amounts to many charities, and given
that individuals give more than a pure free-rider model would predict, we therefore consider how our
predictions change as individuals gain both public and private benefits from giving. Unlike in the
analogous section in part A of this chapter, we will forego another discussion of the Coase Theorem
(which, due to transactions costs, applies only to “small” public goods and only if informational
asymmetries (introduced in Chapter 21) do not impede bargaining) and instead proceed directly
to incorporating a warm glow effect into our model of voluntary giving.
27B.4.1
Public Goods and the “Warm Glow” Effect
Suppose, then, that consumers care about their individual contribution itself – that is, suppose
consumers get a “warm glow” from giving to the public good in addition to knowing that the
overall public good level is higher as a result of their contributions. We could then represent
preferences with the Cobb-Douglas utility function

β γ
α
un (xn , y, zn ) = xα
zn +
n y zn = xn
X
β
zj  znγ ,
(27.51)
max α ln(I − z1 ) + β ln(z1 + (N − 1)z) + γ ln z1 ,
(27.52)
j6=n
where the public good y is simply the sum of all individual contributions. Consumer n’s individual contribution zn therefore enters the utility function twice – once because it contributes to
the overall public good level and once because the individual derives utility from writing a check
for the public good. As the number of consumers increases, the impact of n’s marginal contribution
to y diminishes (giving rise to a worsening free rider problem), but the “warm glow effect” remains
unchanged because it is, in essence, a private good.
Consider a simple example in which there are N consumers that are identical both in their
incomes I and their preferences (that can be represented as in equation (27.51)). Since all individuals
are identical, they will contribute identical amounts z to the public good in equilibrium. Taking
everyone else’s contribution as given, we can then determine how much z1 individual 1 will give to
the public good by solving the problem
z1
where we have incorporated the individual’s budget constraint by expressing x1 = I − z1 and
we have taken the log of the utility function in equation (27.51) to make the derivation of the first
order condition a bit less messy. The first order condition (after re-arranging a few terms) can be
written as
(α + β + γ)z12 + (α + γ)(N − 1)zz1 = (β + γ)Iz1 + γ(N − 1)Iz.
(27.53)
Solving this for z1 would give individual 1’s best response to everyone else giving z to the public
good. But we know that in equilibrium z1 = z – and so we can simply substitute this into the first
order condition and solve for z to get the equilibrium level of contribution by every individual as
z eq =
(β + γN )I
.
β + (α + γ)N
(27.54)
27B. The Mathematics of Public Goods
1095
“Warm Glow” Free
N =1 N =2
y eq
600
1,000
y∗
600
1,200
y eq /y ∗ 1.000 0.833
Riding as Population Increases
N =5 N =10 N =25 N =100
2,059 3,750 8,766
33,775
3,000 6,000 15,000 60,000
0.686 0.625 0.584
0.563
Table 27.3: I = 1, 000, α = 0.4, β = 0.4, γ = 0.2
If you were a social planner choosing z (assuming you constrain yourself to choosing each
individual’s contribution to be the same as everyone else’s), you would set
z∗ =
(β + γ)I
.
α+β+γ
(27.55)
Exercise 27B.18 Verify our derivation of z eq and z ∗ . Then demonstrate that z eq converges to z ∗ as β
goes to zero. Can you make intuitive sense of this?
In Table 27.3, we can then again illustrate how the equilibrium public good level compares to
the optimum as population increases. This is similar to our exercise in Table 27.2 where we assumed
no warm glow from giving and thus simply saw the free rider problem at work. In both cases, we
are setting the exponent on the private good x equal to the exponent on the public good y, but
now we are permitting γ (which was implicitly set to zero in Table 27.2) to be greater than zero to
introduce a warm glow effect. Notice that the previous prediction that free riding will drive private
contributions to zero as population increases now no longer holds because of the private benefit
that individuals get from contributing.
27B.4.2
Marketing Public Goods
Civil society institutions that request voluntary contributions clearly attempt to appeal to the
warm glow that many of us get when we give to a cause we consider worthwhile. Such institutions
may furthermore market their activities in ways that facilitate such a warm glow effect. Consider
our example (from Section A) of an international relief agency that assists poor families in the
developing world. The alleviation of suffering in third world countries is a public good to the extent
that all of us care about it to some extent – and it is a huge public good with huge free rider
problems because it enters so many utility functions. But suppose that the agency can make us
think of our individual contributions to this public good as a private good – by matching us to
specific families that we (and only we, if we believe the marketing) are helping. We can think of
this as the marketing branch of our civil society institution telling us to forget about β in our utility
function and focus on γ. Put differently, in the Cobb-Douglas example we have been working with
where we can think of the exponents as summing to 1, relief agencies – even if they cannot change
how much we care about our own private consumption of x (and thus cannot alter α as a fraction
of the sum of all the exponents) – might be able to persuade us that γ is large relative the β.
How much does this help? Consider the simple example in Table 27.4. Here we assume that
there are 10,000 identical individuals considering a gift to a public good y. We set α = 0.4 and
(β + γ) = 0.6 and then ask how each individual’s gift will change as the share of (β + γ) that is
a “warm glow” increases (i.e. as γ increases relative to β.) The impact is quite dramatic. If each
of us considers our contribution solely to the extent to which it adds to y, we give 15 cents. But
1096
z eq
y eq
Chapter 27. Public Goods
Individual and Total Private Giving with
γ=0
γ=0.1
γ=0.2
γ=0.3
$0.15
$200.08
$333.38
$428.60
$1,500 $2,000,800 $3,333,800 $4,286,000
increasing “Warm Glow”
γ=0.4
γ=0.5
γ=0.6
$500.01
$555.56
$600.00
$5,000,100 $5,555,600 $6,000,000
Table 27.4: I = 1, 000, N = 10, 000, α = 0.4, β + γ = 0.6
if the charitable organization can get us to view even a small portion of what we are giving as a
private good, our contributions go up significantly – and continue going up the more successful
the marketing department in the charitable organization is. The total funding for our charity is
then given in the second row of the table. The “warm glow” effect can therefore help alleviate the
free rider problem by getting individuals to view their contributions as providing both public and
private benefits. However, the effect will never fully overcome the free rider problem – unless we
converge to the extreme case you thought about in exercise 27B.18.
Exercise 27B.19 Suppose the above example applies to a pastor whose congregation has 1,000 members
that get utility from overall donations y to the church as well as their own individual contribution zn . Each
member makes $50,000 and tastes are defined as in equation (27.51) with α = 0.5, β = 0.495 and γ = 0.005.
The pastor needs to raise $1 million for a new church. He can either put his effort into doubling the size
of his congregation, or he can put his energy into fiery sermons to his current congregation – sermons that
will change γ to 0.01 and β to 0.49. Can you show that these will have roughly the same impact on how
much he collects?
27B.4.3
Civil Society and “Tipping Points”
Now suppose that instead of simply deriving some “warm glow” from knowing that we are contributing to a public good, the size of that warm glow is related to how many of our friends are
also giving to the public good. In particular, suppose that the Cobb-Douglas exponent γ depends
on the contribution z by others such that
z
γ(z) = δ1 + δ2 .
(27.56)
I
Plugging this into the first order condition in equation (27.53), we could again solve for the
equilibrium private contribution levels. As you do this, however, you will notice that it has become
more difficult to solve for z eq and that we would have to apply the quadratic formula to solve
eq
eq
for two rather than one solutions – a low zlow
and a high zhigh
.12 Some parameter choices for δ1
and δ2 will make both of these solutions feasible – which implies that we have two different Nash
equilibria. Furthermore, since the equilibrium contributions shape preferences by influencing γ, the
two equilibria result in different preferences depending on which equilibrium we reach.
12 Substituting
γ(z) into equation (27.54) and cross-multiplying, we get
βz + αN z + γ(z)N z = βI + γ(z)N I,
(27.57)
and replacing γ(z) with δ1 + δ2 (z/I), we get (after some more re-arranging of terms)
δ2 N 2
z + (β − (δ2 − α − δ1 )N ) z − (β + δ1 N )I = 0.
I
It is to this expression that the quadratic formula can then be applied.
(27.58)
27B. The Mathematics of Public Goods
eq
zlow
eq
zhigh
1097
Multiple Equilibria when “Warm Glow” is Endogenous
δ2 =0.6
δ2 =0.8
δ2 =1.0
δ2 =1.2 δ2 =1.4
δ2 =1.6
$56.59
$25.57
$16.79
$12.53 $10.00
$8.32
$293.34 $486.88 $593.17 $662.44 $711.40 $747.90
Table 27.5: I = 1, 000, N = 10, 000, α = 0.4, β = 0.4, δ1 = −0.01
In Table 27.5, I calculated the low and high equilibrium contributions for different values of
δ2 just to illustrate how different the multiple equilibria in such settings can be. (The values
of the remaining parameters in the model are reported in the table.) Take the middle column
where δ2 = 1 as an example. In the low contributions equilibrium, we contribute not even 3% of
what we contribute in the high contribution equilibrium! This is because in the low contributions
equilibrium, γ (when α, β and γ are normalized to sum to 1) is 0.0084 – or essentially zero. Thus,
we barely derive a private benefit from giving (because all of us are giving so little) – and we are
essentially just playing the standard free rider game. In the high contributions equilibrium, on the
other hand, the same normalized γ is 0.422, with each of us deriving substantial private benefit
from our public giving.
eq
eq
Exercise 27B.20 * Suppose δ2 = 1. Using δ1 = −0.01 and the values zlow
and zhigh
in the table, derive
the implied level of γ in the two equilibria. (Note that these will not match the ones discussed in the text
because the table does not normalize all exponents in the utility function to sum to 1). Then, using the
eq
eq
parameters for I, N , α and β provided in the table, employ equation (27.54) to verify zlow
as well as zhigh
.
Nothing in the game theory that we have learned makes one of these equilibria more or less
plausible than the other. They are simply two different ways in which individuals might coordinate
their behavior if they in fact value their own contribution to public goods more when their friends are
also contributing. But if a civil society institution finds itself in a “low contribution” equilibrium, it
might find ways to get individuals to coordinate on the “high contribution” equilibrium instead. If
it can get sufficiently many individuals to “temporarily” deviate from their low contribution, then
this makes it more attractive for others to follow suit. The magnitude of the deviations matter a
great deal – because if deviations are not sufficiently large, individuals are likely to fall back into
the “low contributions” equilibrium. But if the institution can induce sufficiently large deviations,
we can cross a “tipping point” where the critical mass has changed their contributions and the
natural tendency is now to fall into the “high contribution” equilibrium.
27B.5
Preference Revelation Mechanisms
As we noted in Section A, individuals typically have an incentive to misrepresent their preferences
for public goods if their contributions to the public good are linked to their stated preferences
for public goods. Economists have therefore thought hard about how to overcome this problem,
and they have proposed “mechanisms” that take into account this incentive problem. The general
study of creating mechanisms that provide individuals with the incentive to truthfully reveal private
information (like their preferences for public goods) is called mechanism design. We will begin by
introducing the general concept and will then illustrate a more general example of a mechanism
(than the one we introduced in Section A) under which individuals reveal their true preferences for
public goods to the institution that requests such information.
1098
27B.5.1
Chapter 27. Public Goods
Mechanism Design
Suppose that A denotes the set of possible outcomes that we may wish to attain, and let {%} denote
the set of possible preferences that individuals might have over these outcomes. For instance, in
the public goods case, A might denote different levels of public goods and different ways of funding
them. An institution like the government might then have in mind some function f : {%}N → A
that would translate the preferences of the N different individuals in the population into the “best”
outcome from A according to some criteria captured by the function f . For instance, in the public
goods case, the government might wish to implement the efficient level of public goods which
depends on the preferences that people in the population have. If the government knew all the
preferences in the population, it could simply do this.
Instead, however, the government needs to request the information about preferences from
individuals in the form of “messages” that individuals can send to the government. Let M denote the
set of possible messages that individuals are allowed to convey to the government. The government
then needs to take all the messages it collects and translate these into an outcome from A; i.e. it
needs to define a function g : M N → A. A mechanism is the combination of the definition of the
types of messages that individuals are permitted to send and the manner in which the messages are
translated into outcomes – i.e. a mechanism is the combination (M, g).
The challenge for the mechanism designer is to define M and g such that the outcome that
emerges from the messages sent by individuals is the same that the government would have chosen
had it simply been able to observe preferences directly and used the function f to pick outcomes.
The mechanism involves “truth telling” if the equilibrium strategy of individuals is to send messages
that truthfully reveal the relevant information about their preferences needed by the government
given that individuals know the function g which the government uses to translate messages into
outcomes. The mechanism is said to implement f if the outcomes that emerge through the application of g to the equilibrium messages sent by individuals are the same outcomes that would
have emerged if f could have been applied directly to the true preferences individuals have. This
is depicted graphically in Graph 27.5 where, rather than being able to directly observe {%}N and
implement f to choose a social outcome from A, a mechanism (M, g) is set up to create a “message
game” in which each player chooses what message to send given that messages are translated to
outcomes through g.
Graph 27.5: Designing a Mechanism
27B. The Mathematics of Public Goods
27B.5.2
1099
The “Groves-Clarke” Mechanism for Public Goods
Suppose then that we consider a world in which N different individuals would benefit from the
provision of a public good y that can be produced at constant marginal cost M C. Our objective
f is to provide the efficient public good level and raise revenues to pay for the cost of doing so.
In order to determine the optimal public good quantity y ∗ , we need to know individual demands
for y, but we typically do not know what these demands are. We therefore need to have the N
individuals report their demands to us by defining a set of possible messages M that they can send
and devise a scheme g by which we are going to settle on a public good level and a payment to
be paid by each of the individuals. The Groves-Clarke mechanism is one such mechanism that has
been proposed.13
The mechanism proceeds as follows, with (1) defining M and (2) and (3) together defining
g : M N → A:
(1) First, individuals are asked to reveal their (inverse) demands for the public good, with each
individual i revealing RDi (y). Such a revealed (inverse) demand curve is depicted in panel (a) of
Graph 27.6 for consumer i. The set of possible messages M is therefore simply the set of possible
downward sloping demand curves.
(2) The institution that implements the mechanism then determines y ∗ as if the revealed demands were in fact people’s actual demands. The RDi curves are thus added up, and y ∗ is set so
that the (vertical) sum of revealed demands is equal to the marginal cost M C of producing the
public good; i.e.
N
X
RDi (y ∗ ) = M C.
(27.59)
i=1
(3) Each individual is assigned a “price” pi in some arbitrary way that has no relation to
what individuals revealed,
P with the only restriction that the sum of the individual pi ’s equals the
marginal costPM C, i.e.
pi = M C. For each individual i, a quantity y i is then defined such that
pi = [M C − j6=i RDj (y)] and the total payment Pi charged to individual i is set to


Z y∗
X
M C −
RDj (y) dy.
(27.60)
Pi (pi ) = pi y i +
yi
j6=i
Graph 27.6 clarifies exactly what the mechanism proposes. In panel (a), we plot the revealed
(inverse) demand curve RDi from consumer iP
– which is a message sent in step (1) above. In panel
(b), we add
to
this
graph
the
curve
(M
C
−
j6=i RDj ). At the intersection of these two curves,
P
(M C − j6=i RDj ) = RDi , which implies that equation (27.59) is satisfied and we have located
y ∗ . Finally, P
in panel (c) we determine the payment owed by consumer i. First, we find where
pi = [M C − j6=i RDj (y)] to define y i . The payment owed by i then consists of the two parts in
equation (27.60): the part pi yP
i is equal to the shaded blue area, while the remainder is the magenta
area underneath the (M C − j6=i RDj (y)) function between yi and y ∗ . The total payment Pi (pi )
owed by consumer i is then simply the sum of the blue and magenta areas.
13 The mechanisms is named for Theodore Groves (1942-) and Edward Clarke (1939-) who separately developed
different versions in the late 1960’s and early 1970’s. William Vickerey (1914-96) is often credited with having hinted
at a similar mechanism in his earlier work on auctions, and some therefore refer to the mechanism as the “VickeryGroves-Clarke mechanism”. Vickery won the Nobel Prize in Economics in 1996 but passed away only 3 days after
the prize was announced.
1100
Chapter 27. Public Goods
Graph 27.6: The Groves-Clarke Mechanism
Graph 27.6c assumes that y i < y ∗ , but it could be that we assigned a high enough pi to individual
i such that the reverse holds. In that case, the integral in equation (27.60) is negative which implies
that consumer i would face a payment that is less than pi y i .
Exercise 27B.21 Illustrate in a graph similar to Graph 27.6 what the payment Pi (pi ) for this individual
would be if pi is sufficiently high such that yi > y ∗ .
27B.5.3
Equilibrium Messages in the Groves-Clarke Mechanism
We can now ask what messages each individual will send in equilibrium under this mechanism.
First, notice the following: The payment Pi (pi ) owed by individual i depends on a number of
variables – none of which except for one can be influenced by the message that is sent by individual
i. To be more precise, the individual has no control over pi which is arbitrarily set by the mechanism
designer. He furthermore has no control over the
P marginal cost M C or the messages RDj (y) sent
by others. Since y i is determined from (M C − j6=i RDj ), he furthermore has no control over yi .
That leaves only y ∗ which is actually affected by individual i’s message! This is key to making the
mechanism work.
Stage 1 of the mechanism – the stage in which individuals send their (inverse) demand curve
messages to the mechanism designer – is a simultaneous move game in which each player settles on
a strategy. We can then ask what consumer i’s best strategy is given what strategies are played
by all other players. And it will turn out that we have defined a simultaneous move game in which
each player in fact has a dominant strategy – i.e. a strategy that is his best response to any and
all messages that others might send.
We can illustrate this by beginning in panel (a) of Graph 27.7 with all the portions of the
problem that are not impacted
P by the message sent by individual i. These are graphed in blue and
include the curve (M C − j6=i RDi ) and the “price” pi assigned to consumer i. We can then add
to this the green (inverse) demand curve that is consumer i’s true demand curve (which only he
knows). If he chooses to tell the truth and reports this as his message, the outcome will be that y t
will be produced, with consumer i charged the shaded area.
27B. The Mathematics of Public Goods
1101
Graph 27.7: Truth telling is Optimal
In panels (b) and (c), we then consider how consumer i will fare if he under- or over-reports his
demand for the public good. Consider first the case where he reports the magenta curve RDiu (y)
is panel (b). The charge he will incur will then be equal to the area (d + e + f ) rather than the
area (b + c + d + e + f ) that he would incur if he told the truth. Thus, by under-reporting his true
demand for the public good, he will save (b+c). But at the same time, his under-reporting will cause
the public good quantity that is produced to fall from y t to y u . If we then use his magenta true
(inverse) demand as his marginal willingness to pay curve,14 we can conclude that this reduction in
the public good will cause him to lose area (a + b + c) in value from the lower public good output.
While he therefore would save (b + c) in payments, he would lose the equivalent of (a+ b + c) in value
from the reduced public good – leaving him worse off by area (a). Under-reporting his demand for
the public good is therefore counterproductive.
In panel (c), we do the analogous exercise for considering whether it might be in the consumer’s
interest to over-report his demand for the public good by reporting RDio . This will increase the
payment he owes from (i + j + k) under truth telling to (g + h + i + j + k) when the consumer
over-reports his demand, thus increasing his payment by (g + h). But the additional value from the
increase in the public good (from y t under truth telling to y o when over-reporting) is only h. Thus,
sending the message RDio rather than the truth results in a loss of (g). Over-reporting is therefore
also counter-productive.
Exercise 27B.22 In Graph 27.7, we considered the case in which yi < y t . Repeat the analysis above
to show that over- and under-reporting is similarly counterproductive when pi is sufficiently high to cause
yi > yt .
Since none of our reasoning has assumed anything about whether individuals other than i are
reporting their demands truthfully, we can conclude that it is in fact a dominant strategy for
14 We know from our consumer theory chapters that uncompensated demand curves can be interpreted as marginal
willingness to pay (or Hicksian) curves only in the case of quasilinear preferences. For simplicity, we are therefore
assuming that underlying preferences are quasilinear. However, while the graphs would get a bit more complex, the
analysis holds also for any set of preferences that are not quasilinear.
1102
Chapter 27. Public Goods
consumer i to report his demand for the public good truthfully. And the same reasoning applies
to all consumers – implying that truth telling is a dominant strategy equilibrium under the GrovesClarke mechanism. This in turn implies that the mechanism will produce the optimal level y ∗ of
the public good.
27B.5.4
Feasibility of the Groves-Clarke Mechanism
While we now know that individuals, when faced with the incentives of the Groves-Clarke mechanism, will report their demands for public goods truthfully, the mechanism will not be feasible
unless it raises sufficient revenues T R for the mechanism designer to actually pay for the total cost
(which is equal to T C = y ∗ (M C) in the absence of fixed costs) of the public good output level y ∗
that emerges. It is easy to illustrate that this is in fact the case.
For each of the individuals affected by the mechanism, one of three scenarios will arise depending
on what pi the individual was assigned: (1) y i < y ∗ , (2) y i = y ∗ or (3) y i > y ∗ . These three cases
are graphed in the three panels of Graph 27.8.
Graph 27.8: Revenues Exceed Costs under the Groves-Clarke Mechanism
In panel (a), y i < y ∗ which results in Pi (pi ) that is equal to the area (a + b + c + d). This area
could be divided into an area pi y ∗ = (a + b + c) plus the remaining shaded triangle (d). In panel
(c), y i > y ∗ which results in Pi (pi ) = (e + f + g + h) – and this area can similarly be divided into
pi y ∗ = (e + f + g) plus the shaded area (h). In both cases, we therefore know that we will collect
(pi ) exactly
pi y ∗ plus some additional revenue. Only in panel (b) where yi = y ∗ is the payment PiP
equal P
to pi y ∗ . The total revenue T R we collect from all consumers is then at least
pi y ∗ , and
since
pi = M C, we can conclude that
TR ≥
N
X
pi y ∗ = y ∗ (M C) = T C.
(27.61)
i=1
We can furthermore see from Graph 27.8 that the only way in which the inequality in the
equation becomes an equality – i.e. the only way that total revenues will exactly equal total costs –
27B. The Mathematics of Public Goods
1103
is if the “prices” happened to be assigned in such a way that y i = y ∗ for all individuals (as illustrated
in panel (b) of the graph). In that special case, the “prices” we have assigned are like real prices
in the sense that individuals pay exactly price times quantity for the public good. In that special
case it is furthermore true that all individuals would in fact choose the optimal public good level y ∗
under the per-unit prices they were assigned. In other words, in that special case, pi is the Lindahl
price for all consumers and we have implemented a Lindahl equilibrium. Of course this could only
happen accidentally under the Groves-Clarke mechanism because the pi ’s are assigned arbitrarily
without knowledge of the underlying demands by individuals.
27B.5.5
A Fundamental Problem in Mechanism Design
Our conclusion that the Groves-Clarke mechanism will almost always raise revenues that exceed
the cost of providing the optimal level of the public good then creates a problem for us: What do
we do with the excess revenue? Remember that we are trying to implement an efficient solution
to the public goods problem – which means that throwing away the excess revenue cannot be the
answer. After all, if we did throw away the excess revenue, we can easily think of a way of making
someone better off without making anyone else worse off: Just give the excess revenue back to
one or some or all of the consumers. But that creates another problem: If we return the excess
revenues, we would create income effects for consumers unless tastes are quasi-linear – which then
would mean that we would alter the optimal level of the public good. Put differently, giving back
the excess revenue alters y ∗ – which means our whole analysis above is thrown out the window. For
this reason, the Groves-Clarke mechanism actually can only implement a Pareto optimum under
the special assumption that individual preferences are quasilinear – a rather strong assumption to
make about preferences we know nothing about at the beginning of the mechanism.
This is a symptom of a much more general problem faced by mechanism designers, a problem
that has become formalized in what is known as the “Gibbard-Satterthwaite Theorem.”15 We will
not develop this formally here, but it bears a striking resemblance to another theorem we will
develop in Section B of Chapter 28. In essence, the theorem says the following: So long as the
f function that the mechanism designer is trying to implement in Graph 27.5 takes into account
the tastes of more than one individual, the function cannot be implemented by any mechanism
that makes truth-telling a dominant strategy unless we can restrict the type of preferences that
individuals have to begin with. In the Groves-Clarke mechanism, for instance, the only way in
which we could implement an efficient outcome was to assume individuals only have quasi-linear
preferences.
The Gibbard-Satterthwaite theorem does leave open the possibility for a mechanism designer
to think up a mechanism that can implement an f function (that takes all preferences into account
and places no a priori restrictions on allowable preference) so long as the designer is content to have
truth-telling emerge as a Nash equilibrium rather than a dominant strategy (Nash) equilibrium.
Thus, it is possible, for instance, to modify the Groves-Clarke mechanism in such a way that there
exists a truth telling Nash equilibrium that results in the optimal provision of public goods with
total revenues exactly equaling total costs. Such mechanisms have in fact been derived, and some
of them are quite simple in terms of the messages they ask consumers to send. Some have even
been implemented in the real world.16
15 The theorem is named for Allan Gibbard (1942-) and Mark Satterthwaite (1945-) who independently developed
the basic result in the early 1970’s.
16 The most famous such mechanism was developed in Groves, Theodore and John Ledyard (1977) “Optimal
1104
Chapter 27. Public Goods
Conclusion
The central problem in public goods provision is found in the existence of positive externalities that
such goods produce and that individuals themselves may not take into account in their consumption
and production choices unless something brings their private incentives in line with the social
goal of efficiency. Without some coordinating device, such individuals are trapped in a Prisoners’
dilemma, each with an incentive to free ride on others, all better off if they could find a way
to enforce cooperation. Still, goods that are, at least to some extent, non-rivalrous are provided
by all sorts of combinations of markets, civil society institutions and governments. When such
goods are excludable, we see them provided in families (among family members), churches, local
communities, competitive firms and clubs. In such settings, individuals find ways of overcoming the
free rider problem and its Prisoners’ dilemma incentives, whether through repeated interactions,
through government subsidies, through Coasian bargaining, through Tiebout competition or by
responding to “warm glow” elements of our tastes. While in some cases the solution is found
solely in voluntary civil society interactions, often such goods are provided through combinations
of markets, civil society and government. As goods become non-excludable and more non-rivalrous,
however, it becomes increasingly difficult to rely on markets or civil society institutions as problems
of free riding and incentives to misrepresent preferences become more intense, and the case for
central government provision of such goods becomes increasingly compelling.
Governments, of course, have their own challenges to overcome. In the case of public goods,
for instance, optimal policy typically requires knowledge of individual preferences that can be
aggregated by the government to determine the appropriate level of public goods. Preference
revelation mechanisms of the type we have discussed in this chapter offer one way to gather such
knowledge, but it has not been one that has, at least thus far, proven terribly practical in most real
world public goods settings. The other natural way in which we attempt to convey our preferences
about public goods is through democratic political processes – processes in which we vote either
directly (or indirectly through our elected representatives) for or against a proposal.
In Chapter 28, we will therefore take on the challenge of thinking about democratic political
processes and the ways in which they gather information on voter preferences and generate policy
outcomes from this information. Since voting is (usually) anonymous, we do not run into the
problem that individuals have an incentive to mis-represent their tastes for public goods – although
we will see that non-anonymous legislators often do have such strategic incentives. In addition we
will see that democratic processes give rise to a whole different set of their own peculiar problems.
End of Chapter Exercises
27.1 We discussed in the text the basic externality problem that we face when we rely on private giving to public
projects. In this exercise, we consider how this changes as the number of people involved increases.
A: Suppose that there are N individuals who consume a public good.
(a) Begin with the best response function in panel (a) of Graph 27.3 – i.e. the best response of one person’s
giving to another person’s giving when N = 2. Draw the 45 degree line into your graph of this best
response function.
(b) Now suppose that all N individuals are the same – just as we assumed the 2 individuals in Graph 27.3
are the same. Given the symmetry of the problem (in terms of everyone being identical), how must the
contributions of each person relate to one another in equilibrium?
Allocation of Public Goods: A Solution to the ‘Free Rider’ Problem,” Econometrica 45, 783-810.
27B. The Mathematics of Public Goods
1105
(c) In your graph, replace y2 – the giving by person 2, with y – and let y be the giving that each person other
than person 1 undertakes (assuming they all give the same amount). As N increases, what happens to
the best response function for person 1? Explain, and relate your answer to the free rider problem.
(d) Given your answers to (b) and (c), what happens to person 1’s equilibrium contribution as N increases?
(Hint: Where on the best response function will the equilibrium contribution lie?)
(e) When N = 2, how much of the overall benefit from his contribution is individual 1 taking into account
as he determines his level of giving? How does this change when N increases to 3 and 4? How does it
change as N gets very large?
(f) What does your answer imply for the level of subsidy s that is necessary to get people to contribute to
the efficient level of the public good as N increases? (Define s as the level of subsidy that will cause a $1
contribution to the public good to cost the individual only $(1 − s).)
(g) Explain how, as N becomes large, the optimal subsidy policy becomes pretty much equivalent to the
government simply providing the public good.
B: In Section 27B.2.2, we considered how two individuals respond to having the government subsidize their
voluntary giving to the production of a public good. Suppose again that individuals have preferences that are
captured by the utility function u(x, y) = xα y (1−α) where x is dollars worth of private consumption and y is
dollars spent on the public good. All individuals have income I, and the public good is financed by private
contributions denoted zn for individual n. The government subsidizes private contributions at a rate of s ≤ 1
and finances this with a tax t on income.
(a) Suppose there are N individuals. What is the efficient level of public good funding?
(b) Since individuals are identical, the Nash equilibrium response to any policy (t, s) will be symmetric – i.e.
all individuals end up giving the same in equilibrium. Suppose all individuals other than n give z. Derive
the best response function zn (t, s, z) for individual n. (As in the text, this is most easily done by defining
n’s optimization as an unconstrained optimization problem with only zn as the choice variable and the
Cobb-Douglas utility function written in log form.)
(c) Use your answer to (b) to derive the equilibrium level of individual private giving z eq (t, s). How does it
vary with N ?
(d) What is the equilibrium quantity of the public good for policy (t, s)?
(e) For the policy (t, s) to result in the optimal level of public good funding, what has to be the relationship
between t and s if the government is to cover the cost of the subsidy with the tax revenues it raises?
(f) Substitute your expression for t from (e) into your answer to (d). Then determine what level of s is
necessary in order for private giving to result in the efficient level of output you determined in (a).
(g) Derive the optimal policy (t∗ , s∗ ) that results in efficient levels of public good provision through voluntary
giving. What is the optimal policy when N = 2? (Your answer should be equal to what we calculated for
the 2-person case in Section 27B.2.2.) What if N = 3 and N = 4?
(h) Can you explain s∗ when N is 2, 3, and 4 in terms of how the externality changes as N increases? Does
s∗ for N = 1 make intuitive sense?
(i) What does this optimal policy converge to as N gets large? Interpret what this means.
27.2
*
In exercise 27.1 we extended our analysis of subsidized voluntary giving from 2 to N people. In the process,
we simply assumed the government would set t to cover its costs – and that individuals would take t as given when
they make their decision on how much to give. We now explore how the strategic setting changes when individuals
predict how their giving will translate into taxes.
A: Consider again the case where N identical people enjoy the public good.
(a) First, suppose N = 2 and suppose the government subsidizes private giving at a rate of s. If individual n
gives yn to the public good, what fraction of the resulting tax to cover the subsidy on his giving will he
have to pay?
(b) Compare the case where the individual does not take the tax effect of his giving into account to the
case where he does. What would you expect to happen to n’s best response function for giving to the
public good in the former case relative to the latter case? In which case would you expect the equilibrium
response to a subsidy s to be greater?
(c) Explain the following true statement: When N = 2, a subsidy s in the case where individuals do not take
the balanced-budget tax consequence of a subsidy into account will have the same impact as a subsidy 2s
in the case where they do.
1106
Chapter 27. Public Goods
(d) Given your answer to (c) (and given that the optimal subsidy level when N = 2 in exercise 27.1 was 0.5),
what do you think s would have to be to achieve the efficient level of the public good now that individuals
think about balanced-budget tax consequences?
(e) Next suppose N is very large. Explain why it is now a good approximation to assume that individual n
takes t as given when he chooses his contribution level to the public good (as he did in exercise 27.1).
(f) True or False: The efficient level of the subsidy is the same when N = 2 as when N is very large if
individuals take into account the tax implication of increasing their giving to the subsidized public good.
(g) Finally, suppose we start with N = 2 and raise N . What happens to the degree to which n’s giving
decisions impact n’s tax obligations as N increases? What happens to the size of the free rider problem
as N increases? In what sense do these introduce offsetting forces as we think about the equilibrium level
of private contributions?
B: Consider the same set-up as in exercise 27.1 but now suppose that each individual assumes the government
will balance its budget and therefore anticipates the impact his giving has on the tax rate t when the subsidy s
is greater than zero.
(a) The problem is again symmetric in the sense that all individuals are the same – so in equilibrium, all
individuals will end up giving the same amount to the public good. Suppose all (N − 1) individuals other
than n give z when the subsidy is s. Express the budget-balancing tax rate as a function of s assuming
person n gives zn while everyone else gives z.
(b) Individual n knows that his after-tax income will be (1 − t)I while his cost of giving zn is (1 − s)zn . Using
your answer from (a), express individual n’s private good consumption as a function of s and zn (given
everyone else gives z.)
(c) Set up the utility maximization problem for individual n to determine his best response giving function
(given that everyone else gives z). Then solve for zn as a function of z and s. (The problem is easiest to
solve if it is set up as an unconstrained optimization problem with only z1 as the choice variable – and
with utility expressed as the log of the Cobb-Douglas functional form.)
(d) Use the fact that zn has to be equal to z in equilibrium to solve for the equilibrium individual contribution
z eq as a function of s. (You should be able to simplify the denominator of your expression to (1 + α(N −
1)(1 − s).)
(e) If everyone gave an equal share of the efficient level of the public good funding, how much would each
person contribute? Use this to derive the optimal level of s. Doest it depend on N ?
(f) True or False: When individuals take into account the tax implications of government subsidized private
giving, the optimal subsidy rate is the same regardless of N – and equal to what it is when N gets large for
the case when people do not consider the impact of subsidized giving on tax rates (as explored in exercise
27.1).
27.3 Everyday Application: Sandwiches, Chess Clubs, Movie Theaters and Fireworks: In the introduction, we
mentioned that, while we often treat public and private goods as distinct concepts, many goods actually lie in
between the extremes because of “crowding”.
A: We can think of the level of crowding as determining the optimal group size for consumption of the good –
with optimal group size in turn locating the good on the continuum between purely private and purely public
goods.
(a) One way to model different types of goods is in terms of the marginal cost and marginal benefit of admitting
additional group members to enjoy the good. Begin by considering a bite of your lunch sandwich. What
is the marginal benefit of admitting a second person to the consumption of this bite? What is therefore
the optimal “group size” – and how does this relate to our conception of the sandwich bite as a private
good?
(b) Next, consider a chess club. Draw a graph with group size N on the horizontal axis and dollars on the
vertical. With additional members, you’ll have to get more chess-boards – with the marginal cost of
additional members plausibly being flat. The marginal benefit of additional members might initially be
increasing, but if the club gets too large, it becomes impersonal and not much fun. Draw the marginal
benefit and marginal cost curves and indicate the optimal group size. In what way is the chess club not
a pure public good?
(c) Consider the same exercise with respect to a movie theater that has N seats (but you could add additional
people by having them sit or stand in the isles). Each customer adds to the mess and thus the cleanup
cost. What might the marginal cost and benefit curves now look like?
27B. The Mathematics of Public Goods
1107
(d) Repeat the exercise for fireworks.
(e) Which of these do you think the market and/or civil society can provide relatively efficiently – and which
might require some government assistance?
(f) Why do you think fireworks on national holidays are usually provided by local governments – but Disney
World is able to put on fireworks every night without government help?
B: Consider in this part of the exercise only crowding on the cost side – with the cost of providing some discrete
public good given by the function c(N ) = F C + αN β with α > 0 and β ≥ 0. Assume throughout that there is
no crowding in consumption of the public good.
(a) Derive the marginal cost of admitting additional customers. In order for there to be crowding in production, how large must β be?
(b) Find the group membership at the lowest point of the average cost function. How does this relate to
optimal group size when group size is sufficiently small for multiple providers to be in the market?
(c) What is the relationship between α, β and F C for purely private goods?
(d) Suppose that the good is a purely public good. What value of α could make this so? If α > 0, what value
of β might make this so?
(e) How does α affect optimal group size? What about F C and β? Interpret your answer.
27.4 Everyday, Business and Policy Application: Competitive Local Public and Club Good Production: In exercise
27.3, we considered some ways in which we can differentiate between goods that lie in between the extremes of pure
private and pure public goods.
A: Consider the case where there is a (recurring) fixed cost F C to producing the public good y – and the
marginal cost of producing the same level of y is increasing in the group size N because of crowding.
(a) Consider again a graph with N – the group size – on the horizontal and dollars on the vertical. Then
graph the average and marginal cost of providing a given level of y as N increases.
(b) Suppose that the lowest point of the average curve you have drawn occurs at N ∗ , with N ∗ greater than
1 but significantly less than the population size. If the good is excludable, what would you expect the
admissions price to be in long run competitive equilibrium if firms (or clubs) that provide the good can
freely enter or exit?
(c) You have so far considered the case of firms producing a given level of y. Suppose next that firms
could choose lower levels of y (smaller swimming pools, schools with larger class sizes, etc.) that carry
lower recurring fixed costs. If people have different demands for y, what would you expect to happen in
equilibrium as firms compete?
(d) Suppose instead that the public good is not excludable in the usual sense – but rather that it is a good
which can be consumed only by those who live within a certain distance of where the good is produced.
(Consider, for instance, a public school.) How does the shape of the average cost curve you have drawn
determine the optimal community size (where communities provide the public good)?
(e) Local communities often use property taxes to finance their public good production. If households of
different types are free to buy houses of different size (and value), why might higher income households
(who buy larger homes) be worried about lower income households “free-riding”?
(f) Many communities impose zoning regulations that require houses and land plots to be of some minimum
size. Can you explain the motivation for such “exclusionary zoning” in light of the concern over free
riding?
(g) If local public goods are such that optimal group size is sufficiently small to result in a very competitive
environment (in which communities compete for residents), how might the practice of exclusionary zoning
result in very homogeneous communities – i.e. in communities where households are very similar to one
another and live in very similar types of houses?
(h) Suppose that a court rules (as real world courts have) that even wealthy communities must set aside
some fraction of their land for “low income housing”. How would you expect the prices of “low income
houses” in relatively wealthy communities (that provide high levels of local public goods) to compare to
the prices of identical houses in low income communities? How would you expect the average income of
those residing in identical low income housing to compare across these different communities?
(i) True or False: The insights above suggest that local community competition might result in efficient
provision of local public goods, but it also raises the “equity” concern that the poor will have less access
to certain local public goods (such as good public schools).
1108
Chapter 27. Public Goods
B: Consider again the cost function c(N ) = F C + αN β with α > 0 and β ≥ 0 (as we did in exercise 27.3).
(a) In the case of competitive firms providing this excludable public good, calculate the long run equilibrium
admission price you would expect to emerge.
(b) Consider a town in which, at any given time, 23,500 people are interested in going to the movies. Suppose
the per auditorium/screen costs of a movie theater are characterized by the functions in this problem,
with F C = 900, α = 0.5, and β = 1.5. Determine the optimal auditorium capacity N ∗ , the equilibrium
price per ticket p∗ and the equilibrium number of movie screens.
(c) Suppose instead that a spatially constrained public good is provided by local communities that fund the
public good production through a property tax. Economic theorists have shown that, if we assume it is
relatively easy to move from one community to another, an equilibrium may not exist unless communities
find a way of excluding those who might attempt to free-ride. Can you explain the intuition for this?
(d) Would the (unconstitutional) practice of being able to set a minimum income level for community members
establish a way for an equilibrium to emerge? How does the practice of exclusionary zoning (as defined
in part A of the exercise) accomplish the same thing?
(e) In the extreme, a model with exclusionary zoning might result in complete self-selection of household
types into communities – with everyone within a community being identical to everyone else. How does
the property tax in this case mimic a per-capita user fee for the public good?
(f)
* Can you argue that, in light of your answer to A(g), the same might be true if zoning regulations are
not uniformly the same within a community?
27.5 * Everyday and Business Application: Raising Money for a Streetlight through a “Subscription Campaign”:
Sometimes, a civil society institution’s goal can be clearly articulated in terms of a dollar value that is needed.
Consider, for instance, the problem you and I face when we want to fund a streetlight on our dark culdesac. We
know the the total cost of the light will be C – and so we know exactly how much money we need to raise. One way
we can raise the money is through what is known as a subscription campaign. Here is how a subscription campaign
would work: We put a money “pledge jar” in between our two houses, and you begin by pledging an amount xY
1 . We
then agree that we will alternate putting a pledge for a contribution into the jar on a daily basis – with me putting
Y
M
in a pledge xM
2 the second day, then you putting in a pledge x3 the third day, me putting in x4 the fourth day,
etc. When enough money is pledged to cover the cost C of the street light, we pay for the light – with you writing a
check equal to the total that you have pledged and me writing a check for the total I have pledged.
A: Suppose you and I each value the light at $1,000 but the light costs $1,750. We are both incredibly impatient
people – with $1 tomorrow valued by us at only $50 cents today. For simplicity, assume the light can be put up
the day it is paid for.
(a) Suppose it ends up taking T days for us to raise enough pledges to fund the light. Let xiT be the last
pledge that is made before we reach the goal. What does subgame perfection imply xiT is? (Hint: Would
it be subgame perfect for person j who pledges the day before to leave an amount to be pledged that is
less than the maximum person i is willing to pledge on day T ?)
(b) Next, consider person j whose turn it is to pledge on day (T − 1). What is xjT −1 ? (Hint: Person j knows
that, unless he gives the amount necessary for i to finish off the required pledges on day T , he will end up
having to give again (an amount equal to what you calculated for xiT ) on day (T + 1) and have the light
delayed by one day.)
(c) Continue working backwards. How many days will it take to collect enough pledges?
(d) How much does each of us have to pay for the streetlight (assuming you go first)?
(e) How much would each of us be willing to pay the government to tax us an amount equal to what we end
up contributing – but to do so today and thus put up the light today?
(f) What is the remaining source of inefficiency in the subscription campaign?
(g) Why might a subscription campaign be a good way for a pastor of a church to raise money for a new
building but not for the American Cancer Association to raise money for funding cancer research?
B: Now consider the more general case where you and I both value the street light at $V , it costs $C, and $1
tomorrow is worth $δ < 1 today. Assume throughout that the equilibrium is subgame perfect.
(a) Suppose, as in A(a), that we will have collected enough pledges on day T when individual i puts in the
last pledge. What is xiT in terms of δ and V ?
(b) What is xjT −1 ? What about xiT −2 ?
27B. The Mathematics of Public Goods
1109
(c) From your answers to (b), can you infer the pledge amount xT −t for t ranging from 1 to (T − 1)?
(d) What is the amount pledged today – i.e. in period 0?
(e) What is the highest that C can be in order for (T + 1) pledges – i.e. pledges starting on day 0 and ending
on day T – to cover the full cost of the light.
P
t
(f) Recalling that ∞
t=0 δ = 1/(1 − δ), what is the greatest amount that a subscription campaign can raise if
it goes on sufficiently long such that we can approximate the period of the campaign as an infinite number
of days?
(g) True or False: A subscription campaign will eventually succeed in raising the necessary funds so long as
it is efficient for us to build the street light.
(h) True or False: In subscription campaigns, we should expect initial pledges to be small – and the campaign
to “show increasing momentum” as time passes, with pledges increasing as we near the goal.
27.6 Business Application: The Marketing Challenge for Social Entrepreneurs: Social entrepreneurs are entrepreneurs
who use their talents to advance social causes that are typically linked to the provision of some type of public good.
Their challenge within the civil society is, in part, to motivate individuals to give sufficient funding to the projects
that are being advanced. Aside from lobbying for government aid, we can think of two general ways in which social
entrepreneurs might succeed in increasing the funding for their organizations. Both involve marketing – one aimed
at increasing the number of individuals who are aware of the public good and thus to increase the donor pool, the
other aimed at persuading people that they get something real out of giving to the cause.
A: We can then think of the social entrepreneur as using his labor as an input into two different single-input
production processes – one aimed at increasing the pool of donors, the other aimed at persuading current donors
of the benefits they get from becoming more engaged.
(a) Suppose that both production processes have decreasing returns to scale. What does this imply for the
marginal revenue product of each production process?
(b) If the social entrepreneur allocates his time optimally, how will his marginal revenue product of labor in
the two production processes be related to one another?
(c) Another way to view the social entrepreneur’s problem is that he has a fixed labor time allotment L that
forms a time budget constraint. Graph such a budget constraint, with ℓ1 – the time allocated to increasing
the donor pool – on the horizontal axis and ℓ2 – the time allocated to persuading existing donors – on the
vertical.
(d) What do the isoquants for the two-input production process look like? Can you interpret these as the
social entrepreneur’s indifference curves?
(e) Illustrate how the social entrepreneur will optimize in this graph. Can you interpret your result as identical
to the one you derived in (b)?
(f) Within the context of our discussion of “warm glow” effects from giving, can you interpret ℓ2 as effort
that goes into persuading individuals that public goods have private benefits?
(g) How might you re-interpret this model as one applying to a politician (or a “political entrepreneur”) who
chooses between allocating campaign resources to mass mailings versus political rallies?
(h) We discussed in the text that sometimes there is a role for “tipping points” in efforts to get individuals
engaged in public causes. If the social entrepreneur attempts to pass such a “tipping point”, how might
his strategy change as the fundraising effort progresses?
B: Suppose that the two production processes introduced in part A are f1 (ℓ1 ) and f2 (ℓ2 ), with dfi /dℓi < 0 for
i = 1, 2 and with “output” in each process defined as “dollars raised”.
(a) Assuming the entrepreneur has L hours to allocate, set up his optimization problem. Can you demonstrate
your conclusion from A(b)?
(b) Suppose f1 (ℓ1 ) = A ln ℓ1 and f2 (ℓ2 ) = B ln ℓ2 with both A and B greater than 0. Derive the optimal ℓ1
and ℓ2 .
(c) In equation (27.54), we determined the individual equilibrium contribution in the presence of a warm
glow effect. Suppose that this represents the equilibrium contribution level for the donors that the social
entrepreneur works with – and suppose I = 1, 000, α = 0.4, β = 0.6. In the absence of any efforts on the
part of the entrepreneur, N = 1000 and γ = 0.01. How much will the entrepreneur raise without putting
in any effort?
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Chapter 27. Public Goods
1/2
1/2
(d) Next, suppose that N (ℓ1 ) = 1000(1 + ℓ1 ) and γ(ℓ2 ) = 0.01(1 + ℓ2 ), and suppose that the entrepreneur
has a total of 1,000 hours to devote to the fundraising effort. Assume that he will in fact devote all 1,000
hours to the effort, with ℓ2 therefore equal to (1000 − ℓ1 ). Create a table with ℓ1 in the first column
ranging from 0 to 1000 in 100 hour increments. Calculate the implied level of ℓ2 , N and γ in the next
tthree columns, and then report the equilibrium level of individual contributions z eq and the equilibrium
overall funds raised y eq in the last two columns. (Obviously this is easiest to do by programming the
problem in a spreadsheet.)
(e) Approximately how would you recommend that the entrepreneur split his time between recruiting more
donors and working with existing donors?
(f) Suppose all the parameters of the problem remain the same except for the following: γ = 0.01(1 + ℓ0.5
2 +
0.001N 1.1 ). By modifying the spreadsheet that you used to create the table in part (d), can you determine
the optimal number of hours the entrepreneur should put into his two fundraising activities now? How
much will he raise?
27.7 Policy Application: Demand for Charities and Tax Deductibility: In end-of-chapter exercise 9.9 of Chapter 9,
we investigated the impact of various U.S. income tax changes on the level of charitable giving. If you have not
already done so, do so now and investigate the different ways that tax policy changes in the U.S. over the past few
decades might have impacted the level of charitable giving.
27.8 Policy Application: Do Anti-Ppoverty Efforts Provide a Public Good?: There are many equity or fairness
based arguments for government engagement in anti-poverty programs – and for general government redistribution
programs. But is there an efficiency case to be made for government programs that redistribute income? One such
possibility lies in viewing government anti-poverty efforts as a public good – but whether or not this is a credible
argument depends on how we think contributions to anti-poverty efforts enter people’s tastes.
A: Suppose there is a set A of individuals that contribute to anti-poverty programs and a different set B of
individuals that receive income transfers from such programs (and suppose that everyone in the population is
in one of these two sets).
(a) In considering whether there is an efficiency case to be made for government intervention in anti-poverty
efforts, do we have to consider the increased welfare of those who receive income transfers?
(b) How would the individuals who give to anti-poverty programs have to view such programs in order for
there to be no externality to private giving?
(c) If your answer to (b) is in fact how individuals view anti-poverty efforts, are anti-poverty efforts efficient
in the absence of government intervention? If the government introduced anti-poverty programs funded
through taxes on those who are privately giving to such efforts already, to what extent would you expect
the government programs to “crowd out” private efforts?
(d) How would individuals have to view their contributions to anti-poverty programs in order for such programs
to be pure public goods?
(e) If the conditions in (d) hold, why is there an efficiency case for government redistribution programs?
(f) If government redistribution programs are funded through taxes on the individuals who are voluntarily giving to anti-poverty programs, why might the government’s program have to be large in order to
accomplish anything?
(g) How does your answer to (f) change if there is a third set of individuals that does not give to anti-poverty
programs, does not benefit from them but would be taxed (together with those who are privately giving
to anti-poverty programs) to finance government redistribution programs.
B: Denote individual n’s private good consumption as xn , the government contribution to anti-poverty efforts
as g and individual n’s contribution to anti-poverty efforts as zn . Let individual n’s tastes be defined as
β γ
un (xn , y, zn ) = xα
n y zn . (Assume that anti-poverty efforts are pure transfers of money to the poor.)
Some argue that private anti-poverty programs are inherently more effective because civil society anti-poverty
programs make use of information that government programs cannot get to. As a result, the argument goes,
civil society anti-poverty efforts achieve a greater increase in welfare for the poor than government redistributive
programs. If this is indeed the case, discuss the tradeoffs this raises as one thinks about optimal government
involvement in anti-poverty efforts.
(a) What has to be true for anti-poverty efforts to be strictly private goods?
(b) What has to be true for anti-poverty efforts to be pure public goods?
27B. The Mathematics of Public Goods
1111
(c) Suppose the condition you derived in (a) applies (and maintain this assumption until you get to part (g)).
Suppose further that there are N individuals that have different income levels – with n’s income denoted
In . Will private anti-poverty efforts be funded efficiently when g = 0? What will be the equilibrium level
of private funding for anti-poverty programs when g = 0 as N gets large?
(d) If the government increases g without raising taxes, will private contributions to anti-poverty efforts
be affected (assuming still that the condition derived in (a) holds)? (Hint: How does the individual’s
optimization problem change?)
(e) Suppose the government instead levies a proportional tax t on all income and uses the funds solely to fund
g. How much private funding for anti-poverty programs will this government intervention crowd out? By
how much will overall contributions to anti-poverty programs (including the government’s contribution)
change? (Consider again the impact on the individual’s optimization problem.)
(f) Can this government intervention in anti-poverty efforts be justified on efficiency grounds?
(g) Suppose instead that the condition you derived in (b) holds. To simplify the analysis, suppose that the
N people who care about anti-poverty programs all have the same income level I (as well as the same
preferences). What is the equilibrium level of funding for anti-poverty programs when g = 0?
(h) What happens to overall funding (both public and private) when the government increases g without
changing taxes?
(i) If the government instead imposes a proportional income tax t and uses the revenues solely to fund g,
what happens to overall funding of anti-poverty efforts assuming the N individuals still give positive
contributions in equilibrium?
(j) Under what condition will the balanced budget (t, g) government program raise the overall funding level
for anti-poverty programs?
27.9 Policy Application: Distortionary Taxes and National Security: In the real world, government provision of
public goods usually entails the use of distortionary taxes to raise the required revenues. Consider the pure public
good “national defense”, a good provided exclusively by the government (with no private contributions).
A: Consider varying degrees of inefficiency in the nation’s tax system.
(a) In our development of the concept of deadweight loss from taxation, we found that the deadweight loss from
taxes tends to increase at a rate k 2 for a k-fold increase in the tax rate. Define the “social marginal cost
of funds” SM CF as the marginal cost society incurs from each additional dollar spent by the government.
What is the shape of the SM CF curve?
(b) True or False: If the public good is defined as “spending on national defense”, then the marginal cost of
providing $1 of increased funding for the public good is $1 under an efficient tax system.
(c) How does the marginal cost of providing this public good change as the tax system becomes more inefficient?
(d) Use your answer to (c) to explain the following statement: “As the inefficiency of the tax system increases,
the optimal level of national defense spending by the government falls.”
(e) What do you think of the following statement: “Nation’s that have devised more efficient tax systems are
more likely to win wars than nations with inefficient tax systems.”
B: Suppose we approximate the demand side for goods by assuming a representative consumer with utility
function u(x, y) = x1/2 y 1/2 and income I, where x is private consumption (in dollars) and y is national defense
spending (in dollars).
(a) If the government can use lump sum taxes to raise revenues, what is the efficient level of national defense
spending?
(b) Next, suppose that the government only has access to inefficient taxes that give rise to deadweight losses.
Specifically, suppose that it employs a tax rate t on income I, with tax revenue equal to T R = tI/(1+βt)2 .
How does this capture the idea of deadweight loss? What would β be if the tax were efficient?
(c) Given that it has to use this tax to fund national defense, derive the efficient tax rate and level of national
defense. (It is easiest to do this by setting up an optimization problem in which t is the only choice
variable, with the utility function converted to logs.) How does it compare to your answer to (a)?
(d) Suppose I = 2, 000. What is national defense spending and the tax rate t when β = 0? How does it
change when β = 0.25? What if β = 1? β = 4? β = 9?
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Chapter 27. Public Goods
(e) Suppose next that the government provides two pure public goods – spending on national defense y1 and
spending on the alleviation of poverty y2 (where the latter is a public good in the ways developed in exercise
(0.5−γ)
27.8). Suppose that the representative consumer’s tastes can be described by u(x, y1 , y2 ) = x0.5 y1γ y2
.
Modify the optimization problem in (c) to one appropriate for this setting – with the government now
choosing both t and the fraction k of tax revenues spent on national defense (versus the fraction (1 − k)
spent on poverty alleviation.)
(f) Does the optimal tax rate differ from what you derived before? What fraction of tax revenues will be
spent on national defense?
27.10 Policy Application: Social Norms and Private Actions: In exercise 21.12 of Chapter 21, we investigated the
role of social norms in determining the number of “green cars” on a city’s streets. Re-visit this exercise and relate
your conclusions to the idea of tipping points from this chapter.
27.11 Policy Application: The Pork Barrel Commons: In representative democracies where legislators represent
geographic districts in legislative bodies (such as the U.S. House of Representatives), we often hear of “pork barrel
spending”. Typically, this refers to special projects that legislators include in bills that pass the legislature – projects
that have direct benefits for the legislator’s district but not outside the district. In this exercise we will think of
these as publicly funded private goods whose benefits are confined to some fraction of residents of the geographical
boundaries of the district. (In exercise 27.12, we will consider the case of different types of local public goods.)
A: Suppose that there are N different legislative districts, each with an equal proportion of the population.
Suppose for simplicity that all citizens are identical – and that tax laws affect all individuals equally. Suppose
further that all projects cost C, and that the total benefits B of a project are entirely contained in the district
in which the project is undertaken.
(a) How much of the cost of a project that is passed by the legislature do the citizens in district i pay?
(b) How much of a benefit do the citizens in district i receive if the project is located in district i? What if it
is not?
(c) Suppose the possible projects that can be brought to district i range in benefits from B = 0 to B = B
where B > C. Which projects should be built in district i if the legislature cares only about efficiency?
(d) Now consider a legislator who represents district i and whose payoff is proportional to the surplus his
district gets from the projects he brings to the district. What projects will this legislator seek to include
in bills that pass the legislature?
(e) If there is only a single district – i.e. if N = 1, is there a difference between your answer to (c) and (d)?
(f) How does the set of inefficient projects that the legislator includes in bills change as N increases?
(g) In what sense do legislator’s have an incentive to propose inefficient projects even though all of their
constituents would be better off if no inefficient projects were located in any district? Can you describe
this as a prisoners’ dilemma? Can you also relate it to the Tragedy of the Commons (where you treat
taxpayer money as the common resource)?
B: Consider the same set of issues modeled slightly differently. Instead of thinking about a number of different
projects per district, suppose there is a single project per district but it can vary in size. Let yi be the size of
a government project in district i. Suppose that the cost of funding a project of size y is c(y) = Ay α where
α > 1, and suppose that the total benefit to the district of such a project is b(y) = By β where β ≤ 1.
(a) What do the conditions α > 1 and β ≤ 1 mean? Do they seem like reasonable assumptions?
(b) Suppose all districts other than district i get projects of size y and district i gets a project of size yi .
Let district i’s legislator get a payoff π i that is some fraction k of the net benefit that citizens within his
district get from all government projects. What is π i (yi , N, y) assuming that the government is paying
for all its projects through a tax system that splits the cost of all projects equally across all districts?
(c) What level of yi will legislator i choose to include in the government budget? Does it matter what y is?
(d) What level of y eq will all legislators request for their districts?
(e) What is the efficient level of y ∗ per district? How does it differ from the equilibrium level?
27.12 Policy Application: Local and National Public Goods as Pork Barrel Projects: Consider again, as in exercise
27.11, the political incentives for legislators that represent districts. In exercise 27.11, we considered pork barrel
projects as publicly funded private goods that residents within the targeted districts enjoyed but everyone paid for.
This resulted in a “Tragedy of the Commons” where legislators view the pool of taxpayer resources as a common
pool that funds their own pet projects for their districts. As a result, such pork barrel projects are over-provided
(much as fishermen overfish publicly owned lakes) – leading to inefficiently high government spending.
27B. The Mathematics of Public Goods
1113
A: Now suppose that the projects in question are not private goods but rather local public goods; that is,
suppose that the benefit B of a project in district i is a benefit that each of the n residents of district i enjoy
equally.
(a) In what way do your answers to A(a) through A(f) of exercise 27.11 change?
(b) Does your basic conclusion from exercise 27.11 still hold?
(c) Next, suppose that each project, while located in one district, benefits all N n citizens of the country
equally; i.e. suppose that projects are national public goods without geographic boundaries in which
benefits are contained. Does your basic conclusion change now?
(d) True or False: The extent to which the fraction of projects requested by legislators is inefficient depends
on the degree to which the benefits of the project are national rather than local.
B: Now consider the way we modeled these issues in part B of exercise 27.11. Each district gets a project,
with the costs and benefits varying with the size of the project. The cost of providing y in a district is again
c(y) = Ay α , but the benefit of the project is reaped by each of the n residents of the district – i.e. the benefit
is b(y) = Bny β . Assume again that α > 1 and β ≤ 1.
(a) Repeat B(b) through B(e) of exercise 27.11 and determine y eq and y ∗ .
(b) Are the projects again inefficiently large? How does the inefficiency vary with N ?
(c) Next, suppose that the benefits of each project are spread across all nN citizens. Derive y eq and y ∗ for
this case of each project being a national public good.
(d) Is there still an inefficiency from having legislators requesting projects for their districts?
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Chapter 27. Public Goods