Question

Suppose there are \(N\) individuals in an economy with three goods. Two of the goods are pure (nonexclusive) public goods, whereas the third is an ordinary private good. a. What conditions must hold for resources to be allocated efficiently between either of the public goods and the private good? b. What conditions must hold for resources to be allocated efficiently between the two public goods?

Short answer

Answer: For efficient allocation between a public good and a private good, the sum of the marginal benefits for the public good for all individuals should equal its marginal cost, and for the private good, the marginal benefit per individual should equal its marginal cost. For efficient allocation between two public goods, the ratio of the sum of the individuals' marginal benefits for each public good should be equal to the ratio of their marginal costs.

Step-by-step solution

(Understanding public & private goods)

(Public goods are non-exclusive and non-rivalrous, meaning that they are available to everyone and one person's consumption does not reduce their availability for others. Examples of public goods are national defense or clean air. On the other hand, private goods are exclusive and rivalrous, meaning that they are only available to those who can pay for them, and one person's consumption reduces their availability for others. An example of a private good is a slice of pizza.)

(Efficient allocation of resources)

(In an economy, resources are allocated efficiently when they maximize the overall utility/welfare of the individuals. In other words, the economy's production should match the individuals' preferences for the goods in question.)

(a. Public good & private good resource allocation)

(For an efficient allocation between a public good and a private good, we need to ensure that the marginal benefit equals the marginal cost for each good. Let \(G_1\) be the public good, \(G_2\) be the private good, and \(C_i\) be the cost of producing the good \(i\). Then the following condition represents the efficient allocation: \(\sum_{n=1}^{N} \frac{\partial U_n}{\partial G_1} = \sum_{n=1}^{N} \frac{\partial C_1}{\partial G_1}\), i.e., the sum of marginal benefits from the public good for all individuals should equal the marginal cost \(\frac{\partial U_n}{\partial G_2} = \frac{\partial C_2}{\partial G_2}\), i.e., the marginal benefit from the private good for each individual should equal the marginal cost)

(b. Public good & public good resource allocation)

(For an efficient allocation between two public goods, we need to ensure that the ratio of the sum of marginal benefits for each good is equal to the ratio of their marginal costs. Let \(G_1\) and \(G_3\) be the two public goods, \(C_1\) and \(C_3\) be the costs of producing them, and \(MB_n^{G1}\) and \(MB_n^{G3}\) be their marginal benefits for individual \(n\). Then, the following condition represents the efficient allocation: \(\frac{\sum_{n=1}^{N} MB_n^{G1}}{\sum_{n=1}^{N} MB_n^{G3}} = \frac{\frac{\partial C_1}{\partial G_1}}{\frac{\partial C_3}{\partial G_3}}\), i.e., the ratio of the sum of individuals' marginal benefits for each public good should be equal to the ratio of their marginal costs.)