Explain how the profit-maximizing rule of setting \(\mathrm{P}=\mathrm{MC}\) leads a perfectly competitive market to be allocatively efficient.

In a perfectly competitive market, firms will produce at the output level where they maximize their profit. This occurs when the marginal cost (MC) of production equals the market price (P) of the good. In other words, a firm will continue to produce more units as long as the additional revenue it earns from selling each unit (marginal revenue) is at least as much as the additional cost of producing it. When MC = P, the firm has no incentive to produce more or fewer units since it would not increase their profit.

Allocative efficiency occurs when the quantity of a good or service produced maximizes the net benefit to society. The net benefit is maximized when the marginal cost (MC) to producers equals the marginal benefit (MB) to consumers. In a perfectly competitive market, the price reflects both the marginal cost and the marginal benefit: P = MC = MB.

In a perfectly competitive market, there are a large number of buyers and sellers, homogeneous products, free entry and exit, and perfect information. These conditions lead to price-taking behavior, meaning both consumers and producers have no influence on market prices. Firms in a perfectly competitive market will produce at the quantity where P = MC, as any deviation from this would lead to a loss of profits.

With P = MC and P = MB, we can deduce that MC = MB in a perfectly competitive market. This relationship implies that resources are allocated in such a way that maximizes net benefit to society. Since the marginal benefit equals the marginal cost, every unit of the good produced adds the same value to society as it does cost to produce. Any change in production would either increase costs without increasing benefits or decrease benefits without decreasing costs, making the market less efficient. In conclusion, the profit-maximizing rule of setting P = MC in a perfectly competitive market ensures that resources are allocated in the most efficient manner. This allocative efficiency is achieved because each unit of the good produced has the same value to consumers as it does cost to producers, and no resources are wasted on producing goods that are not valued by society.