Chapter 33: Public Goods
33.1: Introduction
Some people regard the message of this chapter – that there are problems with the private provision
of public goods – as surprising or depressing. But the message is important nonetheless. As we shall
see, the chapter suggests ways of resolving these difficulties.
33.2: A Simple Public Goods Experiment
I do this experiment from time to time in my lectures. It is potentially costly to me, but I have
sufficient faith in economics and in human nature. You should get your lecturer to run it in his or
her lecture. You could suggest that real money is used – I do that from time to time.
All the students in the lecture room are invited to contribute £10 to a ‘public fund’. A collecting box
will be passed round the room and each student will be asked to put in the box either a £10 note out
of their own pocket or an identically-sized piece of paper on which is written “I do not contribute”
which I have distributed earlier. The point of this is that no-one other than each student knows
whether he or she has contributed £10 or not. After the box has been round the room and all the
students have either put in a £10 note or the slip of paper, it comes to the front of the room and it is
publicly opened. The £10 notes are separated from the slips of paper and the £10 notes counted. I
then take out of my own pocket an identical number of £10 notes and add them to the pile of £10
notes contributed by the students. This now constitutes the ‘public fund’. I then divide up the
resulting public fund equally amongst all the students in the lecture and distribute the proceeds
equally. End of experiment.
Consider for example a lecture with 100 students. Suppose 63 of them contribute a £10 note while
the remaining 37 contribute a worthless piece of paper. When the box is opened and counted a total
of £630 is there. I add another £630 from my own pocket – so we have a grand total of £1260 in the
‘public fund’. This is divided up equally amongst all 100 students present in the lecture – so each of
the 100 students gets £12.60.
As it is important that you understand what is going on, let me give two further examples. In both
we have the same 100 students. In this second example, 35 students contribute a £10 note while the
remaining 65 contribute a worthless piece of paper. When the box is opened and counted a total of
£350 is there. I add another £350 from my own pocket – so we have a grand total of £700 in the
‘public fund’. This is divided up equally amongst all 100 students present in the lecture – so each of
the 100 students gets £7.00. In the third example, 88 students contribute a £10 note while the
remaining 12 contribute a worthless piece of paper. When the box is opened and counted a total of
£880 is there. I add another £880 from my own pocket – so we have a grand total of £1760 in the
‘public fund’. This is divided up equally amongst all 100 students present in the lecture – so each of
the 100 students gets £17.60.
What would you contribute? £10 or nothing?
By this stage in the book you might be beginning to think like an economist. You might argue:
“well no one knows what I am contributing, so I cannot affect what others are contributing. I might
as well therefore take as fixed what others are contributing. Let me suppose that x of the 99 others
are contributing while the rest are not. Then the total from the others is x times £10, and if I
contribute too this will become (x+1) times £10. Both of these totals would be doubled before they
are divided equally between all 100 students. Let me consider my options:
1) If I contribute then I will have to pay £10 and then I will get back 20 (x+1)/100 - a net
return of 20(x+1)/100 – 10
2) If I do not contribute then I pay nothing and get back 20 x/100 - a net return of 20x/100
Which is bigger? 20(x+1)/100 – 10 or 20x/100?
Obviously the second of these, for the first is equal to 20x/100 + (20/100 – 10) and (20/100 – 10)
is clearly negative. The difference between the two is 20/100 – 10 = -£9.80. So contributing the £10
causes you to lose £9.80 – in the sense that you would be £9.80 better off if you did not contribute
irrespective of what anyone else is contributing.
The story is really simple and perhaps is hidden by this bit of maths: if you contribute £10 then you
are already £10 down. What do you get back from this contribution? Obviously the contribution
doubled divided by the number of students in the lecture. This is obviously 20p. You are down on
the deal – by £9.80.
What do you contribute? £10 or nothing?
If you agree with the above reasoning you contribute nothing. You are happy to take your share of
the public fund but realise that you are down on the deal if you yourself make a contribution. If you
contribute then you end up £9.80 worse off than you would do if you did not contribute –
irrespective of what the other students do. To use the terminology of chapter 30, for you not
contributing is a dominant strategy – it is better for you irrespective of what anyone else does.
At the same time, perhaps, you realise that, if everyone thinks and acts the same way as you do, it
will be worse for everyone. If everyone contributed to the public fund everyone would be better off
than if everyone plays their dominant strategy. Let us compare those two extremes: if everyone
plays the dominant strategy then no-one contributes anything and the public fund is zero. I add zero.
Every one gets zero. However if everyone contributes a £10 note then we have a total of £1000 in
the public fund. I then add £1000 – giving a total public fund of £2000. When distributed equally
everyone (except me) walks away with £10 more than at the start of the lecture (the £20 share
minus the £10 they contributed).
All this is very interesting but it does not solve your problem. Suppose you are trying to decide
whether to contribute or not. Consider the two extremes for the other students. We get the following
payoff matrix for you. The entries in the matrix are your payoffs.
You contribute nothing
You contribute £10
All the others
contribute nothing
£0
-£9.80
All the others
contribute £10
£19.80
£10
We notice that in each column the difference between the two rows is £9.80. This confirms what we
have already argued: that you are £9.80 worse off if you contribute than if you do not. But also
notice that there is a minus sign in the bottom left hand cell – you will actually walk away from the
lecture £9.80 worse off if you contribute when the others do not. Moreover - as the top right hand
cell tells us – you will walk away from the lecture with £19.80 extra in your pocket if you do not
contribute while the others do. You are under overwhelming pressure not to contribute.
This is the public good problem. A public good is one that everyone in the public (the lecture) can
enjoy (if they want - they can always refuse to accept the payout). In this experiment the public
good is the share of the public fund – and every one gets it whether they have contributed to it or
not. And the problem is clear – when invited to contribute anonymously to the public good, every
one has a strong incentive not to contribute – to “free-ride” on the contributions of others.
You might argue that we should precede the actual decision by a period of discussion – in which
you point out to the other students the mutual benefits of contributing. You might also want to
include a sort of ‘public declaration’ of what every one intends to do. But, if the actual contributions
are anonymous, this does not change the fact that every one – when it comes to the time of actually
contributing – has a very strong incentive to free-ride by not contributing. You could also institute
some kind of written declaration of intentions, to try to forestall free-riding, but there remain
problems with implementing and verifying such written agreements, particularly when
contributions are anonymous. Perhaps this suggests that things need the intervention of the state in
some form – which seems an eminently sensible suggestion. While one can think of various public
goods that are financed privately it does seem that in practice most public goods are provided
publicly. (The usual examples include all sorts of local amenities like street lighting, policeman,
public parks and libraries, street cleaning facilities, and all sorts of national amenities, particularly
defence. While some of these may actually be done by private firms, the financing of them is
usually done through local and national taxes.)
33.3: All-or-Nothing Public Goods
The example above was of a public good which could be provided in differing amounts, but in
which private contributions were either nothing or some fixed amount. There are other types of
public good – a more familiar one, perhaps, is an all-or-nothing public good. This is one that is
either provided or it is not. Let us at this stage make more clear in what sense it is a public good.
The usual definition is that it is nonrival and nonexclusive. Nonrival means that providing it for one
person provides it for all (in the appropriately defined society). Nonexclusive means that no-one can
be excluded from consuming it (if they want to).
With this kind of public good there are two issues: should it be provided? who should pay for it?
Perhaps economists can say a little on these two issues. We can analyse the key points with a very
simple society in which there are just two individuals A and B. Let us suppose that if the public
good is provided both individuals can consume it (it is nonrival and nonexclusive). Let us suppose
that it costs an amount c to provide it. Whether it should be provided or not depends upon how the
two individuals evaluate it. We can use the concept of the reservation price to decide this. Let us
suppose that individual A’s reservation price for the public good is rA and that individual B’s
reservation price is rB. These are the maximum amounts that the two individuals would pay to
consume the good. In general these reservation prices will depend upon the incomes of the two
individuals – but here we will just take them as given. There are a number of cases to consider:
1)
2)
3)
4)
rA > c and rB > c
rA > c and rB < c or rA < c and rB > c
rA < c and rB < c and rA + rB > c
rA < c and rB < c and rA + rB < c
We can dismiss case 4) immediately: neither individual would be willing to buy the public good
themselves and jointly they do not value it sufficiently to cover the cost. In this case, it would be
clearly inefficient to have the public good. The other cases are more interesting, and in each of these
the provision of the public good could be a Pareto-improvement on its non-provision – depending
upon how the cost is divided between the two individuals. What we mean by a Pareto-improvement
is that both individuals can be better off. In each of these three cases, a division of the cost into a
part paid by individual A, cA, and a part paid by individual B, cB, (where c = cA + cB) such that each
individual pays an amount less than their respective reservation values – that is, cA < rA and cB < rB
– would be possible. In these cases the provision of the public good would be a Paretoimprovement: both individuals would be better off with the good than without it – in the sense that
they are both paying an amount less than their reservation values.
So, in cases 1) through 3) the provision of the public good could be a Pareto-improvement. The
problem is in dividing the cost in an acceptable way – and this depends on what is known about the
reservation values. If both individuals know not only their own reservation value but also that of the
other person, there are a number of possibilities, perhaps relating the cost contributions to the
reservation values1. But it is clear that in some of these cases the individuals have an incentive to lie
about their reservation values. For example, in the first of the two cases under 2), if individual B
knows that individual A values the public good more than its cost, it is obviously in B’s interest to
pretend that his or her reservation value is zero – so that he or she (individual B) ends up paying
nothing while individual A pays the full cost and individual B enjoys the public good for nothing
(that is, he or she free-rides on A). In case 3) – which we could regard as the most empirically
relevant case (particularly when generalised to a society of more than 2 people) – both individuals
have an incentive to lie about their reservation prices (as long as the good is provided of course).
So the problem, both in deciding whether the public good should be provided and in deciding who
should pay for it, reduces to finding out the reservation prices of the members of society. This could
prove difficult – as there is no market in which the reservation prices could be revealed and, as we
have already argued, individuals have a strong incentive to lie about their reservation prices. The
issue then is whether some scheme could be devised which forces individuals to reveal their true
reservation values.
One scheme which we can predict will probably not work is the following. The public good
possibility is announced and all members of society are asked to specify an amount of money which
they are willing to pay towards it. These amounts are added up and if the total exceeds the cost of
the public good then the good is provided and all members of society are sent a bill in which the
cost is proportional to the amount of money that they specified (so that everyone pays an amount
less than or equal to the amount of money they specified). If the total falls short of the cost then the
good is not provided and no-one is billed.
The problem with this scheme is that the cost an individual would pay depends upon the amount of
money they specify. If the good is going to be provided anyhow – (the sum of the amounts specified
by the other members of society already exceeds the cost) then an individual has an incentive to
reduce to zero the amount of money he or she specifies – in this way they end up with the good and
pay nothing. In the case when the individual is pivotal in the decision – that is, when the sum of the
amounts specified by the other members of society falls short of the cost, but would exceed it if his
reservation price were added to the contributions of others - the individual still has an incentive to
reduce the amount specified (to the exact difference between the cost and the sum of the amounts
1
It is difficult for economists to recommend a scheme as the various schemes obviously differ in their distributional
implications.
specified by the others). It seems that in all cases there is an incentive to understate the reservation
price – and to free-ride.
There are schemes which provide an appropriate incentive to state the correct reservation prices,
but these schemes have their drawbacks. For example, the mechanism known as the Clarkes-Grove
mechanism (after the two economists who invented it) gets everyone to reveal their reservation
prices, but it taxes (perhaps rather heavily and certainly unfairly) the individuals who are pivotal in
the decision-making process – that is the ones whose reservation values change the decision from
no provision to provision or vice versa. As a consequence, these schemes, while they might be good
in terms of efficiency, they have drawbacks in terms of distributional aspects. Indeed they seem
politically unattractive. It is interesting to note that it is difficult to find examples of their
application in the real world. In fact, the decision as whether or not to provide public goods, and the
decision as to how they should be financed, seem to be in practice essentially political decisions.
33.4: Variable-Level Public Goods
Other public goods are variable level – in the sense that the level of the provision must be decided.
This is obviously a generalisation of the case considered above. In a sense, the all-or-nothing case is
simple as the question of its provision revolves around the reservation values of the members of
society (though as we have seen, even this case involves difficulties in its implementation). The
variable-level case is more complicated. Let us try and shed some light on it using the tools at our
disposal.
Let us stay with the simple case of a society consisting of just two individuals. Each will have
preferences concerning the public good. Remember that the crucial feature of a public good is that
if it is provided at a certain level, then both members of society can consume it at that level. So the
amount of the public good that both members of society can consume is the sum of the amounts
bought by the two individuals.
Consider the analysis of figure 33.6, in which we do a standard indifference curve analysis of the
optimal choice of the individual. We take a situation in which the individual has to allocate a fixed
monetary income – in this example equal to 50 – between a private good (which only he or she
consumes) and a public good (which both individuals can consume and to which they can both
contribute), both of which have a price of 1 in this example. I assume in this example that the
preferences over the two goods are Cobb-Douglas. In figure 33.6 there are two cases: the left-hand
case in which the other individual contributes nothing to the public good; the right-hand case in
which the other individual contributes 20 to the public good. The variable on the horizontal axis is
the quantity of the public good consumed and that on the vertical axis the quantity of the private
good consumed.
Notice the difference in the two graphs. The large X indicates the endowment point of the
individual. If the other individual contributes nothing to the public good, then this individual has a
budget constraint which goes through the point (0, 50) and has slope equal to –1, minus the relative
prices of the two goods. This is the budget constraint in the left-hand figure. However if the other
individual contributes 20 to the public good then the budget constraint of this individual starts at the
point (20, 50) and has slope –1. This is the budget constraint in the right hand figure. It starts at the
point (20, 50) because this individual, if he or she wanted could consume the 20 units of the public
good that the other individual has supplied and buy 50 units of the private good with the income
that he or she has. (Obviously the budget line does not go to the left of the point X because this
individual can not sell the public good provided by the other individual.)
If we check to see what the individual does in the two situations, we find that in the first case (when
the other individual contributes zero) then this individual spends 35 on the public good and 15 on
the private good2; and in the second case (when the other individual contributes 20 to the public
good) then this individual spends 29 on the public good and 21 on the private good. Note carefully
that in this second case the individual consumes 49 of the public good – of which 20 is provided by
the other individual and 29 by himself3. Thus – when the contribution of the other goes up from 0
to 20 the contribution of this individual goes down from 35 to 29. This is free-riding to a certain
extent – the contribution goes down by 6 when that of the other goes up by 20.
You can probably anticipate the case of complete free-riding – when every increase of 1 in the
contribution of the other causes a decrease in contribution of 1 by this individual. Yes – when the
preferences are quasi-linear.
2
The Cobb-Douglas utility function that I have used has weights 0.7 on the public good and 0.3 on the private good.
In this second case we consider the ‘income’ of the individual to be 70 – his own 50 plus the 20 contribution of the
other.
3
We have the same two cases here – on the left when the other contributes nothing and on the right
when the other contributes 20. You will see that this individual always consumes 50 of the public
good – so every increase in the contribution of the other is always met by a decrease in the
contribution of this individual. In fact, if we graph the contribution of this individual as a function
of the contribution of the other we get figure 33.8. This perhaps best illustrates free-riding
behaviour.
We might be tempted to ask in this situation whether there is a Nash Equilibrium. If we assume that
the two individuals are identical then it is clear that the Nash Equilibrium is where each of the two
individuals is contributing 25 to the public good. (This is the intersection point of the two reaction
curves – where that for one individual is that in figure 33.8.)
But is this an efficient outcome?
We can begin to answer this by pursuing a different line of argument. Suppose each individual
agrees to do the same as the other (on the grounds that they are identical). We could call this the “do
as you would be done by” situation. Then the budget constraint becomes the line illustrated in
figure 33.11.
Where is the best point on this? It is the asterisked point to the right in figure 33.12. This would be
the point chosen if each individual worked on the assumption that the other would contribute the
same as him or her. We can compare that with the Nash Equilibrium – which is also illustrated in
figure 33.12 – the point (50, 25). Which is better? Obviously the “do as you would be done by”
outcome. It leads to a higher indifference curve for both. But what is the problem with this socially
better outcome – yes, it is not on the individuals’ reaction curves and is therefore not an
equilibrium. Does this remind you of anything? It is the prisoner’s dilemma once again. It seems
that private optimising in this public good situation does not lead to the social optimum. Once
again, public intervention seems to be inevitably necessary.
33.5: Summary
A public good is one that can be consumed simultaneously by more than one person – that is
nonrival and nonexclusive.
This chapter points out that there are problems with the private provision of public goods. There are
obvious private incentives for individuals to try and free-ride on the contribution of others.
With all-or-nothing public goods then a necessary and sufficient condition for the optimality of the
provision of the public good is that the sum of the individual reservation values exceeds the cost of
the provision of the public good.
But we saw that there are problems with getting individuals to reveal their true reservation values.
There are mechanisms which might improve things but these have distributional difficulties.
In general with public goods there are clear private incentives for individuals to free ride on the
contributions of others.
The Nash Equilibrium in a variable-level public good game is clearly Pareto inferior to the social
optimum.
Public good provision seems to require political intervention.
This latter conclusion should not surprise us.
33.6: How can an experiment help to understand the Public Goods
problem?
This is a simple experiment which helps us understand the nature of the public goods problem. To
implement it requires a small group of people and some people (‘the experimenters’) who can
control the running of the experiment. The instructions are the following.
The experimenters should organise and implement this public good allocation experiment a
predetermined number of times, which the group as a whole should decide in advance. Each time
the following should be implemented.
All members of society are given an initial endowment of 100 tokens. Each member of society must
individually and simultaneously decide how many of their 100 tokens to ‘put into account A’ and
how many to ‘put into account B’, the sum of these two numbers being less than or equal to 100.
All will declare their decisions individually and simultaneously (the experimenters should decide
how they will collect this information). Then everyone will be told the sum of the amounts put into
account B. Denote this sum by X. Then, as a consequence of these decisions, each member of
society will get paid (in hypothetical money - though all should imagine it to be real) the following
amount in pence:
the amount they themselves put into account A plus the value of X divided by 2
So, imagine that your society consists of 6 people and suppose they put the following amounts into
their accounts A:
0
100
50
20
80
50
50
80
20
50
implying that they respectively put into account B the following:
100
0
implying that the total put into account B was 300. Half of this is 150. Thus the payments received
would be (in pence)
150
250
200
170
230
200
As far as each member of society is concerned they should imagine that they are taking part in an
experiment from which they will take away their earnings over the predetermined number of
repetitions of this experiment. The object is not to beat other ‘subjects’ but to make as much money
as possible.
After playing the experiment, the group should answer the following questions:
(1)
(2)
(3)
(4)
(5)
what has the above experiment to do with Public Goods?
which is the Public Good and which is the Private Good?
what is the Nash Equilibrium in this game?
what is the (best) Pareto Efficient outcome of this game?
what was the outcome in the experiment? Why?
I do not want to give too much away at this stage, but you should note that it is ‘best’ collectively to
put all 100 tokens into account B, whereas, if everyone does what it is best for them personally
(given what the others are doing), then everyone will put nothing into account B. If all 6 people do
the collectively optimal thing, then they will each end up with 300 pence, whereas if all 6 do what is
best for them individually then they will each end up with just 100 pence. There seems a
contradiction here, which you should try and understand. This, of course, is the nature of the public
goods problem.