# Marginal Revenue Marginal Cost and Profit Maximization

Let's begin by looking at the profit-maximizing output decision for any firm, whether the firm operates in a perfectly competitive market or is one that can influence price. Since profit is the difference between (total) revenue and (total) cost, to find the firm's profit-maximizing output level, we must analyze its revenue. Suppose that the firm's output is q, and that it obtains revenue R. This revenue is equal to the price of the product P times the number of units sold: R = Pq. The cost of production C also depends on the level of output. The firm's profit is the difference between revenue and cost:

(Here we show explicitly that tt, R, and C depend on output. Usually we will omit this reminder.)

To maximize profit, the firm selects the output for which the difference between revenue and cost is the greatest. This is shown in Figure 8.1. Revenue R(q) is a curved line, which accounts for the possibility that an increased output may be accomplished only with a lower price. The slope of the line, which shows how much revenue increases when output increases by one unit, is marginal revenue. Because there are fixed and variable costs, C(cf) is not a straight line; its slope,which measures the additional cost associated with an additional unit of output,is marginal cost. C{q) is positive when output is zero because there is a fixed cost in the short run.

For low levels of output, profit is negative-revenue is insufficient to cover fixed .and variable costs. (Profit is negative when q = 0 because of fixed cost.) Here marginal revenue is greater than marginal cost, which tells us that increases in output will increase profit. As output increases,profit eventually becomes positive (for q greater than qo) and increases until output reaches q* units. Here the marginal revenue and marginal cost are equal, and q* is the profit-maximizing output. Note that the vertical distance between revenue and cost, AB, is greatest at this point; equivalently 'it(q)) reaches its peak. Beyond q* units of production, marginal revenue is less than marginal cost, and profit falls, reflecting the rapid increase in the total cost of production.

To see why q* maximizes profit another way, suppose output is less than q *. Then, if the firm increases output slightly, it will generate more revenues than costs. In other words, the marginal revenue (the additional revenue from pro- FIGURE 8.1 Profit Maximization in the Short Run. A firm chooses output q*, so that profit, the difference AB between revenue R and cost C, is maximized. At that output marginal revenue (the slope of the revenue curve) is equal to marginal cost (the slope of the cost curve).

ducing one more unit of output) is greater than marginal cost. Similarly, when output is greater than q*\ marginal revenue is less than the marginal cost. Only when marginal revenue and marginal cost are equal has profit been maximized.

The rule that profit is maximized when marginal revenue is equal to marginal cost holds for all firms, whether competitive or not. This important rule can also be derived algebraically. Profit, tt = R - C is maximized at the point at which an additional increment to output just leaves profit unchanged (i.e., A-n/Aq = 0):

AR/Aq is marginal revenue MR and AC/Aq- is. marginal cost MC. Thus, we conclude that profit is maximized when MR - MC = 0, so that 