Combining Revenue and Costs

In the previous sections in this unit, we analyzed revenue curves. In order to calculate profit, we also need to know the firm’s costs. Using the revenue data and graphs from the previous section and adding typical marginal, average, and average variable cost curves for our magazine firm, we can draw the following graph:

The Profit Maximizing Rule

In any industry, a firm maximizes profits at the point where its marginal cost equals its marginal revenue. Mathematicians use calculus and derivatives to prove this. For an explanation of this proof, see the footnote at the bottom of this page.

Understanding the Profit Maximizing Rule

We can also understand the profit maximizing rule intuitively. In the graph below, Qpm is the profit maximizing quantity. If the firm produces at a point to the left of Qpm (for example, the point at which marginal cost is at its minimum), then we notice that marginal cost is less than marginal revenue. The marginal revenue is $2. The marginal cost at the minimum MC point is at approximately $0.50. By producing this quantity, the firm will add $1.50 to its profit compared to producing the quantity before. Moving a little bit to the right of this quantity, we notice that marginal cost rises to, for example, $1. Shall we produce this quantity over the quantity where MC was $0.50? The answer is yes, because at this higher quantity, we can add an additional $1 to our profit (MR is $2 and MC = $1). Let’s make one more product. The MC rises to, for example, $1.50. Is it profitable to make this additional product? Again, the answer is yes, because we are adding $.50 to our already existing profit (MR is $2 and MC = $1.50). The additional profit is not as high as before, but we are still adding to our total profit. Let’s say that we now produce at Qpm. Here we notice that the marginal revenue of $2 is equal to the marginal cost of $2. Is it profitable to produce this product? Officially, we are indifferent at this point, because we are not gaining or losing any profit. Economists agree that the firm should produce up to or exactly at this quantity.

A quantity point to the right of Qpm shows that the marginal cost exceeds the marginal revenue. For example, if we were to produce at a quantity where the marginal cost is $3, we would lose $1 compared to the profit we had before (MR is $2 and MC = $3). Therefore, we do not want to produce at any point to the right of Qpm. Only at Qpm do we have maximum profits.



The Firm’s Profit and Average Profit

In the graph above, the firm maximizes profits at the quantity where marginal revenue and marginal cost intersect. What is the firm’s profit at this quantity? The graph below illustrates the firm’s profit area. At the profit maximizing quantity, the average revenue is $2. At this quantity, the average total cost equals $1.80. This means that the firm’s average profit is $.20 ($2 minus $1.80). The firm’s total profit is its average profit times the quantity sold. Therefore, total profit equals $.20 times the profit maximizing quantity sold of 100. This equals $20. Total profit is represented by the blue shaded area (length times width of the area).


In the Case of a Loss, but No Shutdown

If the market price decreases (when market demand decreases), then the firm’s demand, average revenue, and marginal revenue curves drop to a lower level. The firm still maximizes its profits at the quantity where MR and MC intersect. However, as average total cost is now above the average revenue curve, it is clear that the firm incurs a loss. Rather than maximizing its profits, the firm is now minimizing its losses. You may ask yourself why the firm will not shut down and produce zero products. In the long run, if the firm continues to incur losses, this will indeed be the case. The firm will go bankrupt. However, in the short run, a firm can incur losses and continue to operate. It will continue to operate as long as the average variable cost is below the price. If the average variable cost is below the price, then the firm’s loss will be less than what it is when it shuts down. When a firm shuts down in the short run, the firm’s loss will be its total fixed costs. As long as the average variable cost is below the firm’s price, then the firm’s loss will be less than its total fixed cost, because the firm makes a profit relative to its variable costs.


In the Case of a Loss and a Shutdown

In the graph below, the price has dropped even further. Now, both average variable and average total costs are above the price. Because the average total cost is above the price, you know that the firm is incurring a loss. Because the average variable cost is above the price, it is now better for the firm to shut down. If the firm does not shut down, its loss will be greater than its fixed cost, because the firm will be losing money on its variable costs. It the firm shuts down, its loss will be its fixed costs. The shutdown may be temporary. If market conditions improve (market demand increases, and therefore, the market price increases), or the firm can manage to lower its costs, then it may become profitable enough for the firm to produce again.


Profit maximizing footnote: In math, a function reaches its maximum (or minimum) when the first derivative of the function equals zero. The profit function is: TP = TR – TC (total profit equals total revenue minus total cost).
The first derivative of the profit function is:
TP’ = (TR – TC)’, which can be rewritten as TP’ = TR’ – TC’.
TR’ is the first derivative of TR. The first derivative deals with changes in its function, so we note that TR’ equals MR. Similarly, TC’ equals MC. 
TP’ = 0 (zero) at the maximum (and minimum) profit level. Thus, MR – MC = 0. This means that MR = MC at maximum and minimum profit levels. In a graph, if the MC curve intersects the MR curve at two different quantities, the lower quantity is the profit minimizing quantity and the higher quantity is the profit maximizing quantity.

Video Explanation
For a video explanation of how to determine the profit maximizing quantity and profit for a firm in pure competition (by looking at a graph), please watch: