Question

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

Short answer

In a perfectly competitive market, a firm is a price taker and can sell any number of units at the market-determined price, making this price equal to the Average Revenue (AR) and the Marginal Revenue (MR). Thus, for a firm in such a market, it holds true that \(P = MR = AR\).

Step-by-step solution

Understanding Perfect Competition

In a perfectly competitive market, firms are price takers. This implies they have no control over the price determined by the market due to the existence of many sellers selling a homogeneous product. The demand curve facing each firm is perfectly elastic, implying any quantity can be sold at the market price.

Understanding Average Revenue (AR)

Average revenue (AR) is the revenue per unit of output sold. It is calculated by dividing total revenue (\(TR\)) by the quantity (\(Q\)). In a perfect competition, AR is equal to the firm's selling price (\(P\)), i.e., \(AR = TR/Q = P\).

Understanding Marginal Revenue (MR)

Marginal revenue (MR) is the additional revenue a firm receives when it sells an additional unit of output. Since the firm in a perfectly competitive market can sell any number of units at the market price, the MR is also equal to the price (\(P\)), i.e., \(MR = P\). As a result, we obtain \(P = MR = AR\).

Key concepts

Price Takers

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)

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)

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.