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Chapter 24: Problem 10

The analysis of public goods in Chapter 24 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 pub lic good? Explain how the characteristics of the public good are reflected in these con ditions? 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 under-provision 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 equi librium guaranteed in this more complex model?

Short Answer

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Short Answer: In the efficient production of a public good for 'n' individuals, the sum of the marginal benefits received by all persons should be equal to the marginal cost of providing the good. It reflects the characteristics of non-rivalry and non-excludability inherent to public goods. Nash equilibrium for 'n' persons equates personal marginal benefits with personal marginal costs and results in under-provision, worsening as the number of persons increase. On the other hand, the Lindahl solution for 'n' persons sets individual cost shares in such a way that the sum equals the total cost of the good. Finding a Lindahl equilibrium for multiple individuals with different preferences can be challenging, but possible under certain conditions.

Step by step solution

01

Define efficient production condition in terms of marginal benefits and costs

For a public good to be produced efficiently, the sum of the marginal benefits received by all n persons should be equal to the marginal cost of providing the public good.
02

Write the condition for efficient production

Mathematically, the condition for efficient production of a public good with n individuals can be written as: \[\sum_{i=1}^{n} MB_i = MC\] where \(MB_i\) is the marginal benefit of the public good to the \(i\)-th individual and \(MC\) is the marginal cost of producing the public good.
03

Explain characteristics of public good

Public goods are characterized by non-rivalry and non-excludability. This means that the consumption of the public good by one individual doesn't reduce the availability for others (non-rivalry) and once the good is produced, no one can be excluded from consuming it (non-excludability). The condition for efficient production captures these characteristics, as it requires that the sum of the benefits received by all individuals should be considered while making production decisions. #b. Nash equilibrium in the provision of public good to n persons#
04

Define Nash equilibrium in the context of public goods

A Nash equilibrium in the provision of public goods occurs when each individual chooses their optimal level of contribution, given the contributions of others. In this situation, no individual has an incentive to change their contribution, given the choices of others.
05

Identify the Nash equilibrium for n individuals

The Nash equilibrium for n individuals can be characterized as a situation where each individual equates their personal marginal benefit of contributing to the public good (denoted as \(MB_i\)) with their personal marginal cost of contributing to the public good (denoted as \(MC_i\)). \[MB_i = MC_i, \forall i \in \{1, 2, ..., n\}\]
06

Explain the inefficiency of the Nash equilibrium

The Nash equilibrium is inefficient because, in this equilibrium, people make their decisions based on their private marginal costs and benefits without considering the benefits others receive from their contributions. Therefore, the sum of the marginal benefits of all individuals is not equal to the marginal cost of providing the public good, which leads to an under-provision of the public good compared to the efficient level.
07

Explain why the under-provision is more severe than in the two-person case

The under-provision of the public good in the Nash equilibrium becomes more severe as the number of persons increases because each person's individual contribution becomes less significant relative to the total cost of providing the public good. This leads to an increased tendency for individuals to "free-ride," resulting in a greater gap between efficient and equilibrium provision levels. #c. Generalization of Lindahl solution to n persons#
08

Define the Lindahl solution in terms of individual shares

The Lindahl solution is an allocation mechanism that determines each individual's share of the cost of providing the public good based on their marginal benefit from the good.
09

Generalize the Lindahl solution to n individuals

The Lindahl solution for n individuals requires determining the cost shares for each individual such that the sum of their contributions equals the total cost of the public good. \[\sum_{i=1}^{n} p_iC(G) = C(G)\] where \(p_i\) is the cost share of the \(i\)-th individual, and \(C(G)\) is the total cost of providing the public good.
10

Discuss the existence of a Lindahl equilibrium in the more complex model

The existence of a Lindahl equilibrium in the more complex model with n individuals is not guaranteed. The challenge in finding a Lindahl equilibrium in this case lies in determining the appropriate cost shares for each individual. This can be particularly difficult when there are many heterogeneous individuals with different preferences, making it challenging to find an allocation that satisfies everyone. Nevertheless, a Lindahl equilibrium may still exist under specific conditions, such as when there is a well-ordered set of individual preferences.

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