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

The condition for efficient production of public goods in an economy with n persons is given by the following equation: $\sum _{i=1}^{n}M{B}_i=MC$, where $M{B}_i$ represents the marginal benefit for person i, and $MC$ represents the marginal cost of producing the public good. This condition is required for efficient production because it ensures that the combined willingness to pay (marginal benefits) of all persons is equal to the marginal cost of producing the public good, taking into account the non-rivalry and non-excludability characteristics of the public good, which ensure that it is accessible to all persons without increasing costs or causing underprovision.

Step-by-step solution

a) Condition for Efficient Production of Public Good

To find the condition for efficient production of the public good for n persons, we need to consider the combined willingness to pay (marginal benefit) for each person's demand for the public good, as well as the marginal cost of producing the public good. This can be expressed as: $\sum _{i=1}^{n}M{B}_i=MC$ Where: - $M{B}_i$ is the marginal benefit for person i - $MC$ is the marginal cost of producing the public good This condition is required for the efficient production of the public good because it ensures that the combined willingness to pay (marginal benefits) of all persons is equal to the marginal cost of producing the public good. The characteristics of the public good, i.e., non-rivalry and non-excludability, are reflected in this condition as they ensure that the public good is accessible to all persons without increasing the cost of production or causing it to be underprovided.

b) Nash Equilibrium and Inefficiency

The Nash equilibrium for the provision of the public good to $n$ persons occurs when each person chooses the optimal level of the public good, given the choices of others. This equilibrium is inefficient because it results in an underprovision of the public good. The main reason for this underprovision is the free-rider problem, where individuals have an incentive not to contribute (or under-contribute) to the public good, hoping that others will cover the cost. The inefficiency of the Nash equilibrium is more severe in the case of $n$-person compared to the two-person case. As the number of individuals in the economy increases, the weight of the individual's contribution decreases, which leads to an increased incentive for an individual to free-ride and rely on others to provide the public good. This causes the overall provision of the public good to be lower than the efficient level.

c) Generalization of Lindahl Solution and Existence

The Lindahl solution is a concept that assigns a personalized price to each of the n persons based on their willingness to pay for the public good. The personalized price reflects the proportion that each individual contributes towards the total cost of providing the public good. The generalized Lindahl solution for an economy with $n$ persons can be expressed as follows: $\sum _{i=1}^n\frac{M{B}_i}{P_i}=MC$ Where: - $P_i$ is the personalized price for person i The existence of a Lindahl equilibrium is not guaranteed in this more complex model. Although the Lindahl solution aims to address the free-rider problem and achieve efficiency, it relies heavily on perfect information about individual preferences and accurate allocation of personalized prices. These requirements can be challenging to fulfill in practice, especially in large economies with a diverse population, leading to the nonexistence of the Lindahl equilibrium.