# The Profit-Maximizing Output

Once a minimum total cost curve is determined, the marginal cost curve can be found from it. Marginal cost is the additional cost of producing one more unit of output or the change in cost divided by the change in output. This should not be confused with average cost, which is total cost divided by total output. Marginal cost plays a leading role in the economist's story of the firm; average cost plays a bit part.

At the risk of repeating material that is becoming stale if you fully understand the logic of the maximization principle, the table below illustrates the logic of finding the level of output at which profit is maximized. Columns 1 and 2 show the total cost curve, and columns 1 and 3 show the demand curve. The total revenue curve comes from the demand curve; it is price multiplied by output. If the firm can sell four units at \$15 each, its total revenue will be \$60.

 Computation of the Best Output for the Firm (1) Output (2) Total Cost (3) Price (4) Total Revenue (5) Which Output (6) Marginal Revenue (7) Marginal Cost 0 \$10 -- -- -- -- 1 20 \$18 \$18 First \$18 \$10 2 31 17 34 Second 16 11 3 43 16 48 Third 14 12 4 56 15 60 Fourth 12 13 5 70 14 70 Fifth 10 14 6 85 13 78 Sixth 8 15

Because marginal revenue is the change in total revenue when another unit is produced and sold, to compute column 6 we simply compute how much the total revenue column changes when output changes. When output increases from 2 to 3 in the table above, total revenue increases from \$34 to \$48, or by \$14. This \$14 is the marginal revenue of the third unit of output. Similarly, because marginal cost is the change in total cost from adding another unit of output, to obtain column 7 we ask by how much the total cost column changes when output changes. When output increases from 2 to 3 in the table above, total cost increases by \$12, from \$31 to \$43. This \$12 is the marginal cost of the third unit of output.

Producing the first unit of output in the table above is worthwhile because it adds \$18 to revenues and \$10 to costs. Paying \$10 to get \$18 is a good deal and the firm should do it. For the same reason the second and third units are worth producing. To produce the fourth unit, the firm must pay \$13 to get \$12. This is not a good deal, so the level of output that maximizes profits is 3. (If it were possible to pick fractional units, it would be worthwhile to go just a bit beyond 3, to the point where marginal revenue exactly equals marginal cost.)

Notice that there is a cost in column 2 when the firm produces zero output. This is a fixed cost, a cost that exists when nothing is produced. For real-world firms fixed cost include rental payments, insurance payments, and depreciation. If fixed costs are high enough, there may be no level of output for which a profit exists. In this case the solution that the maximization principle gives minimizes losses. If losses get big enough and become permanent, the firm may find that the best way to minimize losses is to shut down completely and cease to exist as a firm.

The condition that the firm must set marginal revenue to marginal cost can be expressed in another way. Marginal cost is equal to the change in cost divided by the change in output. Because the equimarginal principle must hold if the firm is on (not above) its total cost curve, the change in cost can come from a change in any of the resources.1 Thus, marginal cost is equal to marginal resource cost divided by marginal product, a change in cost divided by a change in output. Hence, the condition for maximizing profits is

MR = MRC/MP.

In this equation, we can see the role of all three constraints that the firm faces.

There is another way to apply the maximization principle to the firm. Instead of asking how much output it should produce, we can ask how much input should it hire.

1This will be exactly true in the world of calculus, in which changes are so small that they approach zero. It will be only approximately true if changes are larger.