A firm's production process results in a negative externality. At its current profit-maximizing output level ($Q_p$), an analysis of the social welfare function reveals that its first derivative is -10 and its second derivative is -2. What do these values imply about the relationship between the firm's current output ($Q_p$) and the socially optimal output level ($Q_s$)?

Interpreting Calculus for Social Efficiency

Analyzing Social Welfare with Calculus

Consider a scenario where a firm's production generates a negative externality, and the marginal external cost increases with output. At the firm's profit-maximizing output level ($Q_p$), the first derivative of the social utility function is found to be negative. True or False: If the second derivative of the social utility function at this same point ($Q_p$) were positive, it would imply that the socially optimal output level is greater than $Q_p$.

In a situation where a firm's production generates negative external costs that increase with output, we can analyze the social utility function to assess efficiency. Match each mathematical condition related to the social utility function at a specific output level (Q) with its correct economic interpretation.

The Role of the Second Derivative in Efficiency Analysis

In a market with a negative production externality, the social utility function at the profit-maximizing output level ($Q_p$) has a negative first derivative, indicating that social welfare is decreasing at that point. To mathematically confirm that output must be reduced to find the social optimum, the second derivative must also be negative, which signifies that the marginal social utility is ________ as output increases.

An economist is using calculus to construct a formal argument proving that a firm's profit-maximizing output level ($Q_p$) is socially inefficient due to a negative externality. Arrange the following statements into the correct logical sequence that forms this proof.

Critiquing an Economic Analysis of Externalities

A factory's production process creates a negative externality with marginal external costs that increase as output rises. An environmental advocate argues, "At the factory's current profit-maximizing output, social welfare is clearly decreasing. Therefore, any reduction in output from this level will definitively move society closer to the most efficient outcome." Based on a second-derivative analysis of the social utility function, which statement best evaluates the advocate's argument?