Question

Suppose the production possibility frontier for an economy that produces one public good \((P)\) and one private good \((G)\) is given by $$G^{2}+100 P^{2}=5,000$$ This economy is populated by 100 identical individuals, each with a utility function of the form utility \(=\sqrt{G_{i} P}\) where \(G\), is the individual's share of private good production \((=G / 100) .\) Notice that the public good is nonexclusive and that everyone benefits equally from its level of production. a. If the market for \(G\) and \(P\) were perfectly competitive, what levels of those goods would be produced? What would the typical individual's utility be in this situation? b. What are the optimal production levels for \(G\) and \(P\) ? What would the typical individual's utility level be? How should consumption of good \(G\) be taxed to achieve this result? (Hint: The numbers in this problem do not come out evenly, and some approximations should suffice.)

Short answer

b. What are the optimal production levels for goods P and G, what is the utility level of the typical individual, and how could a tax on good G be used to achieve this result?

Step-by-step solution

Set up the problem and rewrite the utility function

Since each individual's share of private good production is \(G_i = \frac{G}{100}\), then we can rewrite the utility function as: $$U =\sqrt{\frac{G}{100} P}$$

Find the points on the PPF where the utility function is maximized

To maximize utility function, we will use the method of Lagrange multipliers with the constraint as the given PPF equation. The maximization problem can be stated as: $$L = \sqrt{\frac{G}{100} P} + \lambda(5,000-G^{2} - 100P^{2})$$ Differentiate with respect to \(G\), \(P\), and \(\lambda\) to find the first-order conditions: $$\frac{\partial L}{\partial G} = \frac{P}{200\sqrt{\frac{G}{100} P}} - 2G\lambda = 0$$ $$\frac{\partial L}{\partial P} = \frac{G}{100\sqrt{\frac{G}{100} P}} - 200P\lambda = 0$$ $$\frac{\partial L}{\partial \lambda} = 5,000-G^{2} - 100P^{2} = 0$$ Solve these equations to find the perfectly competitive and optimal levels of private and public goods production \((G_c, P_c)\) and \((G_o, P_o)\).

Calculate utility levels and the required tax

Using the production levels for perfectly competitive and optimal scenarios, calculate the corresponding utility levels for the typical individual: $$U_c =\sqrt{\frac{G_c}{100} P_c}$$ $$U_o =\sqrt{\frac{G_o}{100} P_o}$$ To find the tax necessary to achieve the optimal result, we need to find the difference in marginal costs between the optimal and competitive scenarios: $$\text{Tax} = \frac{\partial G_o}{\partial P_o} - \frac{\partial G_c}{\partial P_c}$$ To calculate the tax rate, we should first find the optimal and competitive marginal rates of transformation (MRT) by finding the ratio of marginal productivity of the public goods and private goods: $$\text{MRT} = \frac{\partial G}{\partial P}$$ Then, compute the tax rate as the difference of MRTs: $$\text{Tax Rate} = \text{MRT}_{optimal} - \text{MRT}_{competitive}$$ Now we have all the desired values: production levels of \(G\) and \(P\) in a perfectly competitive scenario and their optimal levels, utility levels of the typical individual in both situations, and the tax required to achieve the optimal result.

Key concepts

Utility Function

A utility function helps us understand how individuals make choices based on their preferences. It's a way to measure satisfaction or happiness from consuming goods. In this exercise, the utility function is given by \( U = \sqrt{\frac{G}{100} P} \). This means each individual's utility depends on how much of the private good \( G \) and the public good \( P \) they get. The square root shows diminishing returns; as you consume more, each unit gives you less extra happiness.

In practical terms, utility functions guide how much of a good people want. It's like having a mathematical "shopping list" of preferences aimed at maximizing satisfaction. Since all individuals have identical utility functions here, they share the same preferences towards goods \( G \) and \( P \).

• The private good \( G \) is divided among individuals, so each person gets \( G_i = \frac{G}{100} \).
• The public good \( P \) benefits everyone equally, making its contribution uniform.

Public Good

A public good is one that is non-excludable and non-rivalrous, meaning everyone can use it, and one person's use doesn't reduce its availability to others. In this exercise, the public good \( P \) is produced and enjoyed equally by all 100 individuals in the economy. Whether it's a streetlight or national defense, every citizen benefits from the given level of the public good without additional cost.

This equal enjoyment impacts decisions because no one can be excluded, and it doesn't matter who produces or funds it individually. When a public good is involved, the market alone may not produce the optimal quantity because individuals might rely on others to pay for it. This is known as the free-rider problem, which leads to underproduction in unregulated competitive markets.

• Public goods provide universal benefits, hence why they require more careful consideration in economic planning.
• They often require some form of communal decision-making about how much to provide and fund.

Private Good

A private good is one that is both excludable and rivalrous. An individual’s consumption reduces the amount available for others. In the problem, the private good \( G \) is consumed individually by each of the 100 residents, with each person receiving an equal share \( G_i = \frac{G}{100} \). This division emphasizes our understanding of a private good: everyone gets their own portion.

Private goods are typically allocated by the market based on individual willingness and ability to pay. This can lead to efficiently meeting individuals' specific demands, given prices reflect the costs of production. However, in this scenario, perfectly competitive markets might not yield the best outcomes when public goods are also in play as they disrupt the balance needed to maximize social welfare.

• Private goods help determine utility but must compete for resources against public goods.
• Understanding the nature of private goods allows for personalized consumption preferences in economic models.

Lagrange Multipliers

The method of Lagrange multipliers is a tool used in calculus to find the maxima or minima of a function subject to one or more constraints. In this context, it's applied to maximize the utility function under the constraint given by the production possibility frontier (PPF) \( G^2 + 100P^2 = 5000 \). This is an optimization problem where the goal is to find the best combination of \( G \) and \( P \) that maximizes utility for the 100 individuals.

The Lagrange function \( L = \sqrt{\frac{G}{100} P} + \lambda(5000 - G^2 - 100P^2) \) incorporates both the utility function and the constraint. The derivatives with respect to \( G \), \( P \), and \( \lambda \) provide equations that help solve for the quantities of \( G \) and \( P \) that maximize utility given the constraint.

• This method is vital for understanding how to handle problems where resources are limited.
• By solving these equations, we find the optimal levels of goods that deliver maximum satisfaction under specified limits.

Competitive Markets

Competitive markets are economic systems where multiple producers and consumers interact, setting the price and quantity of goods through supply and demand dynamics. In perfectly competitive markets, like the one hypothesized for \( G \) and \( P \), resources are allocated efficiently such that goods are produced at the lowest cost and distributed based on willingness to pay.

However, when it comes to public goods like \( P \), these markets can fall short because individuals might benefit without paying (free riding), leading to suboptimal outcomes. The market may fail to account for the total benefits of the public good to all individuals, requiring intervention or adjustments, like tax, to reach the optimal production level.

• Perfect competition aims for equilibrium but may need adjustments when public goods are involved.
• The variety and number of market participants influence how well competitive markets function.

Marginal Rate of Transformation

The marginal rate of transformation (MRT) reflects the cost of producing one additional unit of a good compared to another, given resource constraints. In this exercise, the MRT between public and private goods is crucial in understanding how resources shift between these goods on the production possibility frontier (PPF).

It is calculated by the slope of the PPF, reflecting the opportunity cost of increasing production of one good over another. A change in MRT can signal how taxes or subsidies might drive the economy towards optimal production levels. When MRT diverges between competitive and optimal scenarios, it suggests adjustments are needed to align production with social welfare goals.

• MRT is essential for deciding how to allocate resources efficiently between different types of goods.
• Understanding MRT helps in setting economic policies that ensure balanced growth and welfare maximization.