Chapter 33: Public Goods

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. All the others contribute nothing All the others contribute 10 You contribute nothing 0 19.80 You contribute 10-9.80 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 r A and that individual B s reservation price is r B. 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) r A > c and r B > c 2) r A > c and r B < c or r A < c and r B > c 3) r A < c and r B < c and r A + r B > c 4) r A < c and r B < c and r A + r B < 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, c A, and a part paid by individual B, c B, (where c = c A + c B ) such that each individual pays an amount less than their respective reservation values that is, c A < r A and c B < r B 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 values 1. 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 good 2 ; 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 himself 3. 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. 3 In this second case we consider the income of the individual to be 70 his own 50 plus the 20 contribution of the other.

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: implying that they respectively put into account B the following: 0 100 50 20 80 50 100 0 50 80 20 50 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) what has the above experiment to do with Public Goods? (2) which is the Public Good and which is the Private Good? (3) what is the Nash Equilibrium in this game? (4) what is the (best) Pareto Efficient outcome of this game? (5) 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.