Question

The analysis of public goods in this chapter exclusively used a model with only two individuals. The results are readily generalized to \(n\) persons-a generalization pursued in this problem. a. With \(n\) persons in an economy, what is the condition for efficient production of a public good? Explain how the characteristics of the public good are reflected in these conditions. b. What is the Nash equilibrium in the provision of this public good to \(n\) persons? Explain why this equilibrium is inefficient. Also explain why the underprovision of this public good is more severe than in the two-person cases studied in the chapter. c. How is the Lindahl solution generalized to \(n\) persons? Is the existence of a Lindahl equilibrium guaranteed in this more complex model?

Short answer

Answer: The condition for efficient production of a public good in an economy with n individuals is when the sum of marginal benefits equals the marginal cost of production, i.e., \(\sum_{i=1}^{n} MB_i = MC\). The Nash equilibrium is inefficient in this scenario because individuals do not account for the positive externality their contribution brings to others, leading to underprovision of the public good. As the number of individuals increases, the incentive for each individual to free ride (benefit from the public good without contributing) also increases, causing the public good to be even more underprovided.

Step-by-step solution

Recall the efficient production condition

In a two-person economy, the condition for efficient production of a public good is when the sum of marginal benefits equals the marginal cost of production. In a general economy with \(n\) individuals, the same principle applies.

Write the equation for efficient production

Given \(n\) persons, each with their own marginal benefit function of a public good, the condition for efficient production of the public good can be given by: \(\sum_{i=1}^{n} MB_i = MC\) where, \(MB_i\) is the marginal benefit for person \(i\), and \(MC\) is the marginal cost of production. This equation reflects the characteristics of the public good because it indicates that the total benefit of producing the public good, aggregated across all individuals, should equate with the cost of producing the good. #b. Nash equilibrium and inefficiency#

Define Nash equilibrium for public goods

A Nash equilibrium occurs when each individual's chosen contribution to the public good is a best response to the choices made by the other individuals, and no one has an incentive to change their own decision.

Describe inefficiency in the Nash equilibrium

In the Nash equilibrium, each individual does not account for the positive externality their contribution brings to others. This leads to underprovision of the public good, as individuals tend to contribute less than the socially optimal amount.

Explain why inefficiency is more severe in the \(n\)-person case

Compared to the two-person case studied in the chapter, underprovision of the public good is more severe in the \(n\)-person case. This is because the collective action problem becomes more prominent with a larger number of individuals. As the number of people increases, the incentive for each individual to free ride (i.e., benefit from the public good without contributing) also increases, causing the public good to be even more underprovided. #c. Generalization of Lindahl solution and existence of Lindahl equilibrium#

Generalize Lindahl solution for \(n\) persons

The Lindahl solution for \(n\) persons means each person faces a Lindahl tax, calculating their share of the cost of providing the public good, based on their marginal benefit function. In this case, we solve for the set of individual Lindahl prices (\(p_i\)) and the level of public good (\(G\)) that satisfy the efficiency condition: \(\sum_{i=1}^{n} p_i G = C(G)\), where \(C(G)\) represents the total cost of providing the public good. Each person's Lindahl tax rate, \(p_i\), should be set such that their marginal benefit equals their share of the marginal cost: \(MB_i(G)=p_i MC(G)\) for each \(i\), \(i= 1, 2, ... , n\)

Discuss the existence of Lindahl equilibrium in the \(n\)-person model

The existence of a Lindahl equilibrium in this more complex model depends on the particular structure of the individuals' preferences and the cost functions. If a stable point can be found where all individuals' marginal benefits equal their share of the marginal costs, and the efficient condition is satisfied, then the Lindahl equilibrium exists. However, there is no guarantee that such an equilibrium can always be found, as the solution may depend on the specific characteristics of the public good, cost functions, and individuals' preferences.

Key concepts

Efficient Production Condition

When discussing public goods, the notion of production efficiency is crucial. In microeconomics, a public good, such as a national defense system or public parks, is non-excludable and non-rivalrous, meaning it is available to all, and one person's consumption does not reduce its availability to others.

To achieve efficiency in the production of a public good—where the resources are utilized in the best possible manner to maximize community satisfaction—economists refer to an essential condition: the sum of each individual’s marginal benefits (\(MB_i\) for individual 'i') from the public good should equal the marginal cost (\(MC\)) of producing that good. This is represented mathematically as \[\sum_{i=1}^{n} MB_i = MC\] for an economy with 'n' individuals. This efficient production condition reflects the collective nature of public goods, as it ensures that the good is produced until the last unit where the total value to the society justifies the production cost.

Understanding this concept is vital when analyzing collective decisions about resource allocation for public goods. It indicates a socially optimum production level, guiding policymakers in public spending.

Nash Equilibrium in Public Goods

The Nash equilibrium is a fundamental concept in game theory, illuminating how individuals behave in strategic situations. When applied to public goods, it uncovers a standard predicament. In a Nash equilibrium for public goods provision, every person chooses the amount to contribute based on the others' contributions, leading to a stable state where no one wishes to change their contribution unilaterally.

However, this scenario presents a flaw: inefficiency. The equilibrium tends to be inefficient because it results in the underprovision of the public good. That's because individuals often ignore the full value their contribution brings to others—the positive externality—and thus contribute less than what would be socially optimal.

The inefficiency escalates as more people join the group (\(n\) persons). This is due to the 'free rider' problem, where individuals have less incentive to contribute, expecting others to bear the cost. This collective interaction aggravates the dilemma, causing a more significant shortfall in the provision of the public good compared to a two-person scenario.

Lindahl Solution for Public Goods

The Lindahl solution offers a theoretically equitable method for financing public goods. It suggests that people should pay for the public good in proportion to the benefit they derive from it. For 'n' individuals, this implies a personalized price or tax for each person, aligning with their marginal benefit from the good.

Under this method, the goal is to find the set of Lindahl prices (\(p_i\) for individual 'i') that fulfill the following two conditions:

• Every individual's marginal benefit equals their share of the marginal cost (\(MB_i(G) = p_i MC(G)\)).
• The sum of the individual payments equal the total cost of providing the public good (\[\sum_{i=1}^{n} p_i G = C(G)\]), where \(C(G)\) stands for the total cost.

The Lindahl solution, elegant in theory, faces practical challenges. The existence of a Lindahl equilibrium isn't guaranteed in a more complicated multi-person context, as it hinges on individuals accurately revealing their valuation for the public good, which can be strategically manipulated. Moreover, finding a stable point where everyone's marginal benefits equal their share of the costs can be arduous, given the diversity of preference and cost function complexities in a community.