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

Allocative efficiency occurs when resources are allocated in a way that maximizes the net benefit to society. In other words, an allocatively efficient market produces the quantity of goods and services that is most desired by consumers at the lowest possible cost. This is achieved when the Marginal Benefit (MB) of consuming an additional unit of a good or service is equal to the Marginal Cost (MC) of producing that unit.

In a perfectly competitive market, firms seek to maximize their profits. One way to achieve this is by setting the price of their products equal to the marginal cost of production (P = MC). This means that the additional revenue gained from selling one more unit of a good or service (Marginal Revenue, or MR) must be equal to the additional cost of producing that unit (MC).

In a perfectly competitive market, there are many buyers and sellers, and each buyer and seller is a price-taker, meaning they cannot influence the market price. The market price is determined by the interaction of supply and demand. As a result, individual firms (sellers) face a horizontal demand curve at the market price, meaning their Marginal Revenue equals the market price (P = MR). For buyers (consumers), the market price represents the marginal benefit they receive from consuming an additional unit of a good or service (P = MB).

To show that a perfectly competitive market is allocatively efficient, we need to prove that the profit-maximizing rule leads to a situation in which MB = MC: 1. Firms maximize their profits by setting P = MC (Profit-maximizing rule) 2. In a perfectly competitive market, P = MR for sellers and P = MB for buyers 3. Therefore, by transitivity (if a=b and b=c, then a=c), we have: MR = MC (because P = MC) MR = MB (because P = MR) 4. Given that MR = MC = MB, we reach the conclusion that a perfectly competitive market is allocatively efficient. This means that resources are allocated in the most desirable way, maximizing the net benefit to society. In summary, the profit-maximizing rule of setting P = MC leads a perfectly competitive market to be allocatively efficient because it ensures that the production and consumption of goods and services occur at a level where the marginal benefit to consumers is equal to the marginal cost of production. This optimal allocation of resources maximizes the overall welfare of society.