Chapter 5: Public goods

Chapter summary

When a public good is provided, it can be enjoyed simultaneously by all consumers. Such simultaneous consumption violates the assumption of the private nature of goods underlying the efficiency of a competitive equilibrium. If there are public goods in the economy, market failure occurs and the unregulated competitive equilibrium will fail to be efficient. This inefficiency implies that there is a potential role for government intervention.

The chapter begins by defining a public good and distinguishing between public goods and private goods. This provides an insight into why the market failure occurs. The inefficiency is then demonstrated by analysing the equilibrium that is achieved when consumers are responsible for purchasing a public good. Following this, the Samuelson rule characterising the efficient provision of a public good is derived. This permits a comparison of equilibrium and optimum. The focus then turns to consideration of methods through which efficiency can be achieved.

The first of these, the Lindahl equilibrium, is based on the observation that the price each consumer pays for the public good should reflect their valuation of it. The Lindahl equilibrium achieves efficiency but, since the valuations are private information, it generates incentives for consumers to provide false information. Mechanisms designed to elicit the correct statement of these valuations are then considered. The chapter is completed by a discussion of the outcomes of experiments designed to test the extent of false statement of valuations and of how market data can be employed to calculate valuations.

5.1 Public and private goods

A public good can be distinguished from a private good by the fact that it can provide benefits to a number of users simultaneously whereas a private good can, at any time, only benefit a single user. If the public good can accommodate any number of users without a decrease in quality then it is said to be pure. It is impure when congestion occurs and quality falls as the number of users increases.

A pure public good has the following two properties.

• Non-excludability: If the public good is supplied, no consumer can be excluded from consuming it except, possibly, at infinite cost.
• Non-rivalry: Consumption of the public good by one consumer does not reduce the quantity available for consumption by any other.

In practice, it is difficult to find any good that satisfies perfectly both the conditions of non-excludability and non-rivalry.

To obtain further insight into these definitions it is helpful to think of a continuum of types of good running from purely private goods, for which there is complete rivalry and exclusion at zero cost, to pure public goods.

5.2 Market failure

Public goods may be made available either through the purchases of individual consumers or through provision by the government (or by a combination of both). This section demonstrates that relying entirely upon individual purchases will result in inefficiency.

Assume there are two consumers who have incomes M^1 and M^2 . Each consumer’s income must be divided between purchases of a private good and a public good. The preferences of consumer h are described by the utility function

U^h = U^h (X^h, G) (5.1)

where x is consumption of the private good and G is the total quantity of the public good... For consumer 1 substituting for the quantity of private good their utility can be written as

U^1 (x^1,G) = U^1 (M^1-g^1, g^1 + g^2) (5.2)

The decision problem of consumer 1 is to choose g^1 to maximise the utility level.

The essential economic feature of private purchase of public goods: there is strategic interaction between the consumers. What is meant by this is that the utility level of consumer 1 depends upon the choice of g 2 made by consumer 2. This is because the public good purchased by consumer 2 benefits consumer 1 because of the non-excludability and non-rivalry.

Activity Example

Assume each dollar spent on the private good gives you 10 units of utility but each dollar spent on the public good gives you and your two neighbours 5 units each. How much would you spend on the public good? What is value of total purchases at the Nash equilibrium? If you and your neighbours had $10 each, what level of expenditure on the public good maximises the total level of utility?

5.3 Efficient provision

It has been shown that private provision will lead to an equilibrium from which a Pareto improvement can be made by a simultaneous increase in public good purchases by both consumers. The intention now is to provide a description of the Pareto-efficient allocations. The result that will be derived is called the Samuelson rule in honour of its discoverer.

The marginal social benefit of an extra unit is found by summing the marginal benefits of the consumers. For efficiency, this sum of marginal benefits must be equated to the marginal cost.

5.4 Personalised prices

The first method of achieving efficiency involves using an extended pricing mechanism with prices ‘personalised’ to capture the valuation that each consumer places upon the public good.

This idea can be further understood by considering the differences between public and private goods. With private goods, consumers face a common price but choose to purchase different quantities according to their preferences. In contrast, with public goods, all consumers must consume the same quantity. This can only be efficient if the consumers choose to purchase the given quantity of the public good. They can be induced to do so by correctly choosing the price they face.

The idea of personalised pricing can be captured by considering the government announcing the share of the cost of provision of the public good that each consumer must bear. For example, it may say that each of the two consumers must pay half the cost of the public good. Having heard the announcement of these shares, the consumers then state how much of the public good they wish to have supplied. If they both wish to have the same level, then that level is supplied. If their wishes differ, the shares are adjusted and the process repeated. The adjustment continues until a pair
of shares is reached at which they both wish to have the same quantity. This final point is called a Lindahl equilibrium.

The reason why efficiency is attained can be seen in the illustration of the Lindahl equilibrium. A typical indifference curve of consumer 1 is denoted I 1 and of consumer 2, I 2 . The shape of these reflects the fact that each consumer prefers more of the public good but dislikes an increased tax rate. The locus of the vertical points of the indifference curves for a consumer is called the Lindahl reaction curve. These reaction curves plot the ideal level of the public good for
each consumer as a function of the tax shares.

5.5 Misrevelation

Although personalised prices seem a very simple way of resolving the public good problem, when considered more closely a number of difficulties arise in actually applying them. Firstly, there is the very practical problem of determining the prices in an economy with many consumers. The practical difficulties involved in announcing and adjusting the individual shares are essentially insurmountable. Secondly, there are also issues raised concerning the incentives for consumers to reveal their true demands.

By announcing preferences that do not coincide with their true preferences, it is possible for a consumer to shift the outcome in their favour, provided that the other does not do likewise. False announcements are an example of the general problem of misrevelation of preferences.

If a consumer acts strategically, they are able to manipulate the outcome to their advantage. This suggests that the search for a means of attaining the Samuelson rule should be restricted to allocation mechanisms that cannot be manipulated in this way.

5.6 Mechanism design

Consumers have an incentive to reveal false demand information when personalised prices are being determined, which follows from the consistent application of the assumption of utility maximisation..

The decision facing the players is to choose either to produce or not produce a fixed quantity of a public good. If the public good is not produced then G = 0. If it is produced, G = 1. The cost of the public good is given by C = 1. The gross benefit of the public good for players 1 and 2 is given by v 1 = v 2 = 1. Since the social benefit of providing the good is v 1 + v 2 = 2 which is greater than the cost, it is socially beneficial to provide the public good.

In equilibrium both players will understate their valuation of the public good. As a result the public good is not provided despite it being socially beneficial to do so. The reason is that the proportional cost-sharing rule gives an incentive to under-report preferences for the public good. With both players under-reporting, the public good is not provided.

Each consumer knows the benefit they will gain if the public good is provided and they know the cost they will have to pay. The difference between the benefit and the cost is called the net benefit. This can be positive or negative. The decision rule is that the public good is provided if the sum of reported net benefits is (weakly) positive.

The special feature of the Clarke-Groves mechanism is the structure of the payoffs. If the public good is not provided, each consumer receives a payoff of 0. If the good is provided, then each consumer receives a side payment equal to the reported net benefit of the other consumer.

The problem with this mechanism is the side payments that have to be made. If the public good is provided and v 1 = v 2 = +1 then the total side payments are equal to 2 – which amounts to the total net benefit of the public good. These side payments are money that has to be put into the system to support the telling of truth. Obtaining the truth is possible, but it is costly.

5.7 Obtaining valuations

The challenges involved in designing a preference revelation mechanism to obtain a consumer’s valuation of a public good have been noted. Despite these results, there are still reasons for believing that in reality the situation is not as bad as it appears. There are two major reasons for this. The first is that consumers may not act as strategically as the theory suggests. If they do not, then they may be willing to reveal the truth even if it is not in their interests to do so. The second reason is that consumers often reveal their valuation of a public good indirectly via observed market purchases. When they do this, there is no need for the use of a mechanism.

As a practical method of determining public good valuations, it has been suggested that two preference revelation mechanisms should be run simultaneously. The first would be designed to lead to under-reporting of the true valuation of the project and the second to over-reporting.

There are also market-based methods for deriving valuations. When acting in a competitive market, a consumer has no incentive to reveal false information about their preferences. Thus compare a house (market good) in different environments. The difference in price of two houses that are identical in all respects except for environmental quality captures the value of the environmental difference through inference.

5.8 More on private provision

A Pareto improvement could be made from the initial equilibrium point if both consumers simultaneously raised the quantity of public good purchased. This was sufficient for developing a contrast with efficient provision and for investigating mechanism design. As well as predicting inefficiency, the model of private provision generates several more strong results.

Income distribution invariance, is a consequence of the fact that the utility levels of the consumers are linked via the quantity of public good.

The private provision model therefore implies that, when the consumers have identical utilities, contribution behaviour will equalise utilities even in the face of income differentials.

The important point is what happens to the equilibrium level of provision as the number of consumer tends to infinity (the idealisation of a ‘large’ population). What happens can be seen by considering the consequence the equilibrium will be at the point where the reaction function crosses the vertical axis. As this point is reached, the provision of each consumer will tend to zero but aggregate
provision (the infinite sum of many zeros) will be positive. This result can be summarised by saying that in a large population each consumer will effectively contribute nothing. The result can be further extended to show that in a large population only the richest consumer will ever contribute.

The conclusion that only the richest consumer in a large population will contribute is not an accurate representation of, for example, charitable contributions in the USA. Nor does the average level of contribution appear to be close to zero. At the private provision equilibrium an increase in contribution by one consumer will lead to a reduction by all others. This feature has also been criticised as an inaccurate representation of reality.

5.9 Experimental evidence

The analysis of private provision demonstrated that the equilibrium will not be Pareto efficient and that, compared to Pareto-improving allocations, too little of the public good will be supplied. A simple explanation of this result can be given in terms of each consumer relying on others to contribute and hence deciding to contribute little themselves. In a sense, each consumer is free-riding on others’ contributions and, since all attempt to free-ride, the total contribution fails to reach an efficient level.

The basic structure of the experiments is to give participants a number of tokens that can be invested in either an individual exchange or a group exchange. These experiments do not provide great support for the equilibrium based on the private provision economy with Nash behaviour. In the
single-period games free-riding is unambiguously rejected. What seems to be occurring is that the participants are initially guided more by a sense of fairness than by Nash behaviour. When this fairness is not rewarded, the tendency is then to move towards the Nash equilibrium.

Moving to non-Nash conjectures can therefore alter the equilibrium level of the public good but unfortunately does not eliminate the invariance properties. The major objection to this approach is that it is entirely arbitrary. There are sensible game-theoretic motives for focusing upon the Nash equilibrium and these are not matched by any other set of conjectures.

A final modification is to remove the individualism and allow for social interaction by modifying the rules of social behaviour. In the same way that this can arise in tax evasion, it can occur with public goods. One way to do this is to introduce reciprocity under which each consumer considers the contributions of others and contrasts them to what they feel they should make.

Reading: Jean Hindriks and Gareth D. Myles (2004), Intermediate Public Economics

Chapter 8 Public Goods

"If there are goods such as national defense in the economy, market failure occurs and the unregulated competitive equilibrium will fail to be efficient. This inefficiency implies that there is a potential role for government intervention." p137

"The pure public good has been the subject of most of the economic analysis of public goods. In some ways, the pure public good is an abstraction that is adopted to provide a benchmark case against which other, more realistic, cases can be assessed. A pure public good has the following two properties.

• Non-excludability. If the public good is supplied, no consumer can be excluded from consuming it.
• Non-rivalous. Consumption of the public good by one consumer does not reduce the quantity available for consumption by any other"

"Public goods which are excludable, but at a cost, or suffer from congestion beyond some level of use are called impure. The properties of impure public goods place them between the two extremes of private goods and pure public goods." p138

"Club goods are public goods for which exclusion is possible." p139

"Public goods do not conform to the assumptions required for a competitive economy to be efficient. Their characteristics of non-excludability and non-rivalry lead to the wrong incentives for consumers. Since they can share in consumption, each consumer has an incentive to rely on others to make purchases of the public good. This reliance on others to purchase is call *free-riding* and it is this that leads to inefficiency." p139

"The Nash equilibrium is therefore not Pareto efficient although it is privately efficient. No further Pareto improvements can be made when a point is reached where the indifference curves are tangential." p143

"Efficiency in consumption for private goods is guaranteed by each consumer equating their marginal rate of substitution to the price ratio. The strategic interaction inherent with public goods does not ensure such equality." p143

"However, this does not imply equality of the marginal rates of substitution because the indifference curves are defined over different combinations of public good purchased by the two consumers. As will soon be shown, the efficiency condition involves the sum of marginal rates of substitution and is termed the Samuelson rule in honour of its discoverer." p143-144

"... the marginal rate of substitution should be viewed as a measure of the marginal benefit of another unit of the public good. The marginal cost of a unit of public good is one unit of private good. Therefore the rule says that an efficient allocation is achieved when the total marginal benefit of another unit of the public good, which is the sum of the individual benefits, is equal to the marginal cost of another unit. The rule can easily be extended to incorporate additional consumers: the total benefit remains the sum of the individual benefits.

Further insight into the Samuelson rule can be obtained by contrasting it with the corresponding rule for efficient provision of two private goods. For two consumers, 1 and 2, and two private goods this is

MRS1 = MRS2 = MRT" p144

"The consequence of these observations is that efficiency will not be attained through direct public good provision with the use of any of the forms of taxation discussed so far. This finding provides the motivation for considering alternative allocation mechanisms that can provide the correct level of public good by eliciting preferences from consumers." p145

"So that the Median Voter Theorem of Chapter 4 can be applied, assume that there is an odd number, H, of consumers where H > 2 and that each of the consumers has single-peaked preferences for the public good. This second assumption implies that when the level of utility is graphed against the quantity of public good there will be a single value of G h that maximizes utility for consumer h. Such preferences are illustrated in the lower panel of Figure 8.6. The consumers are numbered so that their preferred levels of public good satisfy G 1 < G 2 < ... < G H ." p146