Explain why it is true that for a firm in a perfectly competitive market, \(P=M R=A R\).

In a perfectly competitive market, individual firms operate as 'price takers.' This means that they accept the market price as given and have no influence over it. Why is this the case? Imagine a marketplace filled with numerous sellers, each offering an identical product. Since there are so many sellers and the product is exactly the same no matter from whom you buy, none of these sellers can charge more than the current market price without losing their customers to competitors.

The demand curve for a price taker is perfectly elastic. In other words, a firm can sell as much as it wants at the market price, but if it tries to increase its price even slightly, the quantity demanded for its product will drop to zero. This scenario enforces the firm's status as a price taker and ensures that the strategic individual pricing decisions are not applicable. Understanding this is crucial when dissecting the connection between price, average revenue (AR), and marginal revenue (MR) in a perfectly competitive market.

Average revenue (AR) simplifies the understanding of how much a firm typically earns per unit of the product sold. It's a straightforward but essential concept in identifying the performance of a business in the context of sales and pricing strategies. Calculated by dividing the total revenue (TR) by the quantity sold (Q), it comes down to this simple equation:
\[ AR = \frac{TR}{Q} \]
In a perfect competition, where price takers reign, the AR is equal to the selling price (P). The product's uniform price across all sellers means that as long as the firm sells at the market price, the average revenue per unit remains constant, regardless of how many units are sold. Therefore, in such markets, \( AR = P \), signaling that the average revenue does not change with output – a distinctive trait of perfect competition.

Marginal revenue (MR) is a concept that might seem puzzling at first, but it is the lifeblood of understanding firm behaviors in economics. MR represents the additional income obtained from selling one more unit. Technically, it's the derivative of the total revenue function with respect to the quantity. But in a perfectly competitive market, where each additional unit of output can be sold at the constant market price (P), the marginal revenue doesn't change with each additional unit sold. This results in a formula that is simple yet profound:
\( MR = \frac{\Delta TR}{\Delta Q} = P \)
It details that the extra revenue from an extra unit of a product will be exactly the price at which the product is sold. Since price takers in such a market face a horizontal demand curve, all the outputs are sold at this same price leading to the conclusion \( P = MR \). Connecting this back to the average revenue, for a firm in perfect competition, it follows that \( AR = MR = P \), painting a clear picture of how intimately price, AR, and MR are linked in this market structure.