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

Let's first understand some key terms: - Allocative Efficiency: It occurs when goods and services are produced and distributed in such a way that maximizes overall social welfare. - Perfect Competition: A market structure in which there are many firms producing identical products with no barriers to entry or exit. Firms in a perfectly competitive market are price takers and have no individual control over the price. - Price (P): The amount a consumer pays for a good or service. - Marginal Cost (MC): The additional cost incurred by producing one more unit of a good or service.

In a perfectly competitive market, firms aim to maximize their profits. The profit-maximizing rule states that a firm will maximize its profits when it sets the price (P) equal to the marginal cost (MC) of producing an additional unit of the good or service. Mathematically, it can be represented as: \[P = MC\]

Under allocative efficiency, the total surplus (sum of consumer and producer surplus) in the market is maximized. Consumer surplus represents the difference between what consumers are willing to pay and what they actually pay. Producer surplus represents the difference between the price firms receive and the minimum amount they would be willing to accept (their marginal cost) to produce a good. When Price equals Marginal Cost (\(P = MC\)), it implies that resources are allocated efficiently, and the value that consumers place on the good (willingness to pay) equals the cost of producing the good. There is no deadweight loss, meaning that the total surplus in the market is maximized.

Perfect competition pushes firms to produce at the lowest possible cost and motivates them to only produce goods and services that consumers value at or above the cost of production. When firms follow the profit-maximizing rule (\(P = MC\)), they allocate resources efficiently because they only produce goods that generate a positive total surplus. In summary, the profit-maximizing rule, the rule of setting Price equal to Marginal Cost, leads a perfectly competitive market to be allocatively efficient because it maximizes the total surplus (sum of consumer and producer surplus) in the market. This efficient allocation of resources ensures that goods and services are produced and distributed in a way that maximizes overall social welfare.