by Filip Palda
Samuelson’s rule
If we can overcome the prisoner’s dilemma, we must then ask how much of a public good to provide. In the case of private goods, this is not a question we need to ponder because the private market figures out this level according to the second welfare theorem. In the case of a non-excludable good, as public goods are, private property rights are not feasible, so neither is a market in which individuals reveal their preferences and costs of production and reach deals that no one else can beat—the very essence of Pareto-efficiency. In the case of public goods, government needs some idea of preferences and costs. But public goods are also non-rivalrous, and this adds an additional layer of difficulty in figuring out the efficient level of public good. Paul Samuelson solved the puzzle of reaching this Pareto-efficient level in his famous four-page missive, The Pure Theory of Public Expenditure. He applied the logic of market equilibrium for rivalrous private goods, to non-rivalrous goods, and discovered that it was grossly Pareto-inefficient. He then borrowed from the Hicks-Kaldor compensation principle to reassure us that his rule had the potential to be Pareto-efficient. This may all seem like a debate about how many angels can fit on the head of a pin, but as we will see, this was the debate that helped unite the science of economics with politics, and helped to clarify what the proper limits to government action should be in Pareto’s Republic.
The logic of Samuelson’s rule is simple. Keep increasing the level of a public good as long as the sum of what people are willing to pay exceeds the cost of providing that increase. To see this, consider a public singer visiting a village of ten residents where everyone listens to the performance. The service is non-rivalrous because one person’s enjoyment does not detract from that of any other. For this singer’s service, each villager might be willing to contribute a bowl of soup, although the singer would be happy with just one. Clearly, some level of payment could be found to improve everyone’s well-being, so there is room for improvement. A similar, second singer willing to accept only one bowl of soup could accompany the first, but now the villagers would only be willing to contribute half a bowl of soup each, as the novelty of performance has started to fade. This is a Pareto-improving addition, but there is still room to improve both the lot of singers and villagers. Singers should be brought in until no villager is willing to contribute more than a tenth of a bowl of soup to the last one. Bring in a further singer beyond this previous one and the sum of what people are willing to pay would fall below one bowl of soup and no Pareto-improving exchange would be possible. Once no Pareto improvements are possible, you have attained Pareto efficiency.
And that is the Samuelson rule for achieving the Pareto-efficient level of a non-rivalrous good or service in its simplest form. Just keep increasing the level while the sum that all people are willing to contribute together is still more than the cost of providing that unit. Stop increasing the non-rivalrous good or service at the point where joint willingness to pay is equal to the extra cost of provision.
I say that is the rule in its simplest form because a wrinkle seems to arise when villagers differ in their willingness to pay but are all forced to contribute the same amount of soup. With varying degrees of willingness to pay, Samuelson’s rule could easily violate Pareto efficiency. If you followed Samuelson’s rule, you could run into a situation where one villager was willing to pay half a bowl of soup less than the admission price and another was willing to pay a full bowl of soup more. In this case, the rule would say keep increasing the number of singers because the sum of extra willingness to pay exceeds the extra cost by a half, even though one villager would clearly suffer, and by his or her suffering, strip Samuelson’s rule of its Pareto efficiency. The way to turn this seeming violation of Pareto efficiency into a win-win situation is to have the more enthusiastic villager compensate the less enthusiastic villager for his loss. He can still perform this compensation and come out ahead because his gain from the increase is bigger than the other villager’s loss.
All that matters for Samuelson’s rule to work is that some change to the supply of a public good be such that the sum of what some people are willing to pay above their tax price for the increase exceeds what some other people would have to be compensated because their tax “price” is superior to what they are willing to pay. Basically, if some benefit greatly, the windfall can be spread to others. The possibility of compensating losers is known as the Hicks-Kaldor compensation principle and this is what allows Samuelson’s rule to lead us towards the Pareto-efficient provision of a non-rivalrous good or service.
Compensation is also at the heart of Pareto efficiency in private markets for rivalrous goods and services. In that case, the search for Pareto efficiency can be left to individuals because rivalrous goods are privately consumed. Each individual can seek out others who can compensate him or her with a cash payment in exchange for the good and need not take into account the use of this good on others, because rivalrous goods do not directly affect anyone else’s well-being in the way that non-rivalrous goods do. The good passes from hand to hand until it finds the person willing to offer the greatest compensation for its use. No such individual process is possible for non-rivalrous, non-excludable goods, but the hope among economists is that Hicks-Kaldor compensation could be the practical way of making Samuelson’s rule work in the real world. All it would take is information on what people are willing to pay.
The informational challenge to Samuelson’s rule
Private markets expose what people are willing to pay, and what producers can charge if pushed to the wall by competition from rivals. Government does not have access to such an information revelation mechanism, as Hayek pointed out, because it cannot exploit the increasing returns to the private search for information open solely to the individual and revealed only in a competitive market situation. Put more plainly, you have to basically be in the situation to understand what the value of a reallocation of resources is.
No power, however august or highly placed can out-think the man or woman on the street buying purse straps or remaindered books because experience is knowledge. Imagine a spy-satellite filming two men talking. What are they saying? Did they pass between them a book or a cheque? More importantly, what do both feel is the value of continuing their meetings and exchanges? This is true in the case of rivalrous goods and services, which we examined at length in the previous chapter, and it is also cripplingly true in the case of non-rivalrous goods and services. Yet whereas government could help a society attain Pareto efficiency in the case of rivalrous resources by creating and protecting private property rights, it has no such a possibility when it comes to non-rivalrous goods which happen also to be non-excludable.
Non-excludable, non-rivalrous resources are called public goods and cannot be supplied by the private market due to the free-rider problem even though there is a need for them. It seems we are stuck with the need for government to provide these goods, and even though we know the correct formula, courtesy of Paul Samuelson, we are missing one very important piece of information: willingness to pay. In the market for private, rivalrous goods, this information comes out through the competitive process. In the case of a public good there is no competitive process because due to the lack of excludability there is no market; consequently, there is no readily available measure of willingness to pay, which means that government faces a major problem in implementing the Hicks-Kaldor compensation criterion, the basis for the practical Pareto-efficient implementation of the Samuelson rule.
Knowing what your customers are willing to pay is an ongoing challenge to private entrepreneurs. Those that get the calculation right thrive. Those that miscalculate disappear from the market. Economist Armen Alchian suggested that trial and error by entrepreneurs is part of an evolutionary process that leads private markets to Pareto efficiency in a sort of contest to determine who is fittest to survive. Entrepreneurial ventures in private markets are dying and being born all the time. Their stories provide surviving entrepreneurs
with constantly updated lessons on the needs of the market. It is this process that helps push society to Pareto efficiency. No such stories are available for public goods because there are no markets, only the stories professors of economics tell. Is there no solution to this problem?
The problem with implementing the public goods rule is that government has trouble determining what people want the way a market does in order to divine how much they wish to pay. Government must learn about the needs of people through elections, or in the case of an autocratic regime, through the murmuring of courtiers and the din of rioting in the streets. Even elections can be an unreliable guide to public good demand because of the separation of the act of voting from the obligation to pay for what one is getting. Government agents might ask people directly, but the response might not be well thought-out because nothing is directly at stake for the person. The answer could be exaggerated if the person thinks others will pay most of the tax, or understated if the person thinks a disproportionate amount of the tax will fall on him or her. There is no getting around this challenge. Governments cannot gather information continually as do markets because governments are big. They change seldom by comparison with firms. So gathering accurate information is a problem.
Economists and political scientists agonized for decades over the question of how government might get to know what people were willing to pay for public goods. Looking directly at private markets to see how much people were willing to pay for certain goods was not much help because private markets failed to produce or price public goods such as street lighting, national security, a legal system, and disease control. The data were simply not there.
Some hope for pricing public goods came from the work of economist Sherwin Rosen. In the 1970s, Rosen invented “hedonic analysis,” which, under very strict theoretical conditions, could take something like the price of a house, and unravel it to show how street lighting, parks, and other public amenities contributed to its value. By isolating how much of the benefit of a public good seeps into housing value, hedonic analysis could indirectly measure the value of public goods. Yet the method was also plagued by lack of relevant data on housing characteristics and some very strict theoretical conditions were needed to make it a practical tool. It was a hot topic in economics in the 1970s but seems to not have had much impact on the science of pricing government services, perhaps because of all the “ifs” involved. Later in the 1970s, Swiss researchers, most notably Bruno Frey and Werner Pommerehne, thought they had come upon a solution to the problem of pricing public goods. They argued that one could draw inferences about willingness to pay in the political market in the same way as one could from the private market. Look at what voters are willing to pay in taxes for different levels of public goods, they said. If you find similar communities but with different tax levels and different levels of public goods consumption, you might be able to build some notion of the relation between what people were willing to pay in taxes for a given level consumed.
It was a business of connecting the dots. The first dot was the community with low taxes and low public goods. The second dot was the community with slightly higher taxes and more public goods. By connecting the dots you could establish how much people were willing to pay for a certain level of public good. Once you had this information on willingness to pay, you could compare it to the costs of providing the public good, and keep providing that good as long as the costs were less than the willingness to pay. Here, then, seemed to be a heuristic, or rule-of-thumb for supplying the Pareto-optimal level of a public good.
The statistical picture these researchers painted was clouded by the possibility that the link between the level of public goods supplied and what people were paying was only a surface impression of linkage and not a causal relation. Perhaps people paid more for higher levels of public goods because some unaccounted force, such as a rise in wealth, had given communities the ability to ask for more. There are statistical techniques for filtering out or “controlling” for the effect of a third factor interfering with causal interpretations, but these techniques rely for their effectiveness on assumptions that are so tenuous that they can be thought of as exercises in divination. Yet the more fundamental objection to this approach was the simplest. How do you know that the governments in the other communities got the calculation right? If they did not provide the correct level of public goods for a given tax price, you were basing your government decisions upon the mistaken premises of other governments.
The first Queen Elizabeth said, “I have no desire to make windows into men’s souls.” Yet what else can economists do but peep into souls in order to prescribe optimal levels of public goods? Or so it seemed until, gradually, researchers began to understand the link between Samuelson’s formula and the work of Duncan Black that has since come to be called the “median voter theorem.”
The median voter model in relation to Samuelson
One way out of the problem of using community level data, or hedonic analysis, to divine demand for public goods was to build a model that could predict who decides policies in a democracy. The most popular model so far has been that of the median voter, which Duncan Black produced in a 1948 article entitled, On the Rationale of Group Decision-making. Black’s model marks the start of the modern mathematical era of the study of government and excellent reading on this can be found in The Encyclopedia of Public Choice. Black’s work also formed a bridge between elections and questions of the Pareto-efficient supply of public goods, and for this reason it should be considered the foundational research piece that united politics and economics under the banner of Pareto-efficient thinking.
Black’s theory says that if the debate over public goods is simple enough to be put on a left-to-right scale, such as would be the case with military, or health expenditures, and if a few other highly technical assumptions about people’s preferences hold, then the level of public good a government provides will be that which is preferred by the median voter. No one has ever met this voter, but in theory, he or she sits in the middle of the desired level of public good of all other voters, which means that slightly fewer than fifty percent want more than the median voter, and slight fewer than fifty percent want less. The median voter decides elections.
A politician may propose a level of a public good which is greater than what three-quarters of voters want. A rival could win the election by proposing a level which is greater than what only two-thirds of voters want. He would win because his position is closer to what most people want than is the position of the first candidate. Both candidates know this game and gravitate to the position of the voter who sits in the middle of the crowd. If for each person willing to pay a certain amount less than the median voter there is another person willing to pay the same amount more, then the distribution of preferences around the median is said to be symmetric. With symmetric distributions, the median also turns out to be the average. To see this, consider three people. The first is willing to pay $1 for the public good, the next willing to pay $2, and the third willing to pay $3. The average of these three amounts is $2, which is precisely the median voter’s willingness to pay.
This long chain of reasoning leads to a profound insight. If you multiply the tax price for the public good which results from the median voter’s preferences by the number of people, you get the sum of the tax dollars people are willing to pay for the public good at the level actually supplied. Provided the tax price per unit of public good reflects the cost of producing an additional unit, then the election results in a government that follows Samuelson’s rule. Put more simply, if the median voter represents the average voter, then you get public goods geared toward the average person. By a mathematical quirk, that average can also coincide with Samuelson’s rule for the optimal provision of a public good.
Finding that the median voter’s preferences can lead to the Pareto-efficient amount of public good being produced seems to do away with government’s need to worry about calculating the voter’s de
mand for public goods. All that is needed to produce the right amount of public good is taxation and a competitive election, just as competition and property rights were the only ingredients needed for markets to produce the Pareto-efficient level of private goods. Never before had intellectuals forged such a sharp argument for justifying government intervention as they had with Samuelson’s rule grafted to Black’s median voter theorem.
Despite the elegance of Samuelson and Black’s analysis, we should not allow ourselves to apply their insights without also understanding the limitations of these insights. The Samuelson rule only applies to public goods, that is, projects that have the potential to serve all people in a non-rivalrous manner. These are the types of government intervention that unite people by working in the interests of all. Samuelson’s rule does not apply to the selfish redistribution of income such as might happen when a cynical interest group gets a law passed that protects it from competition and allows it to raise prices at the expense of consumers. Nor does it preserve any link with the median voter theorem when property rights are not protected by law. In a democracy that has no property protection, voting can become the means by which some groups rob the resources of other groups through taxation and redistribution. This is the exact opposite of the peace of Pareto. For us to have faith that Samuelson’s rule can work through the medium of the median voter theorem, some economic prerequisites to democracy need to be present. We shall see what these are two chapters from now, but you may not be surprised to know that the most important prerequisite is the protection of property rights.
The tragedy of the commons
Public goods lead to too little private initiative to produce them because their non-excludable character means that people can free ride on the service provided. With too many free riders and too few paying customers, such a service would fail. There is an opposite problem. Sometimes people provide not too little, but too much effort. In the case of public goods, people free ride, or under-participate in providing the good because they benefit without paying the costs. In the opposite problem, people provide too much effort because they benefit from their effort without taking into account the cost this imposes on others. This is the diametrically opposite, but logically close relative of public goods, and it is called the problem of the common property resource.