# The Profit-Maximizing Level of an Input

The determination of the profit-maximizing level of an input is, like the determination of the profit-maximizing level of output, a mathematical problem if there is perfect knowledge of the supply curves of resources, the production function, and the demand curve. It is another application of the maximization principle, which says that the best level of an input is that level for which its marginal benefit to the firm--the extra money the firm can obtain by hiring or buying the input--just equals the marginal cost to the firm of hiring or buying the input. In the jargon of economists, the marginal revenue product of an input should equal the marginal resource cost.

The marginal resource cost depends only on the supply curve of the input. The marginal revenue product is the extra revenue that the firm can obtain from hiring another unit of the resource. It depends both on the extra output that the input produces and on the extra revenue the firm can obtain from each extra unit produced. For example, if adding another unit of an input can increase output by three, and selling an extra unit of output increases revenues by \$4.00, the marginal revenue product is \$12.00. If it costs an extra \$10.00 to buy or hire the input, the firm will increase profit by \$2.00 if it buys or hires it. If the input costs an extra \$14.00, the firm will decrease profits by \$2.00 by using it, and should consider a reduction in the level of this particular input.

Because marginal revenue product equals the marginal product of a resource multiplied by the marginal revenue of the output, the condition that marginal resource cost should equal marginal revenue product, or:

MRC = MRP

can be written as

MRC = MP x MR = Marginal Product x Marginal Revenue

This formula can, with a bit of algebraic manipulation, be turned into the condition for profit-maximizing output, or:

MR = MC = MRC/MP

A simple numerical example shows the equivalence of the two approaches--by finding optimal output or optimal input--to maximizing profits and the role of the three constraints. Assume that a firm is a price taker both as a seller of output and as a buyer of inputs, and that it can sell as much as it wants at a price of \$2.10 per unit and buy as much labor as it wants for \$10.00 per unit. Assume also that the amount of capital is fixed at two, that capital and labor are the only inputs, and that the firm faces the production function discussed earlier.

 Computation of the Best Input Level for the Firm (1) Marginal Revenue Product (2) Marginal Resource Cost (3) Labor (4) Output (5) Marginal Product (6) Marginal Revenue (7) Marginal Cost -- -- 0 0 -- -- -- \$27.30 \$10.00 1 (first) 13 (th) 13 \$2.10 \$.77 14.70 10.00 2 (second) 20 (th) 7 2.10 1.43 12.60 10.00 3 (third) 26 (th) 6 2.10 1.67 10.50 10.00 4 (fourth) 31 (st) 5 2.10 2.00 8.40 10.00 5 (fifth) 35 (th) 4 2.10 2.50

Columns 3 and 4 of this table show data taken from the production function. Column 5 shows the marginal product of labor, derived from columns 3 and 4. Column 2 shows the marginal resource cost of labor, which comes from the assumption that the firm can buy as much labor as it wants at \$10.00 per unit. Column 6 shows marginal revenue, which comes from the assumption that the firm can sell as much as it wants at \$2.10 per unit of output.

To find the profit-maximizing amount of labor, the firm must compare the extra cost of another unit of labor with the extra revenue that the extra labor adds. The extra cost is the marginal resource cost, shown in column 2. The extra revenue is the marginal revenue product, the value to the firm of the extra output that the additional labor produces. Column 1 shows marginal revenue product, which is the product of marginal revenue and marginal product (columns 5 and 6). By comparing the marginal revenue product to the marginal resource cost, one can immediately see that the profit-maximizing amount of labor is four. Hiring the fourth unit of labor adds \$10.50 to revenue and \$10.00 to cost, so profits increase. Hiring a fifth unit adds less to revenue than to costs, \$8.40 and \$10.00 respectively, so it is not a profitable unit to hire. If the firm hires four units of labor, the production function says that it will produce 31 units of output.

The previous section explained that to calculate profit-maximizing output, one needs to compare marginal cost of output with marginal revenue. Column 6 of the table contains the marginal revenue. To find the marginal cost of output, one must compute what it costs to produce another unit. When the firm hires the first unit of labor, it adds 13 to output and \$10.00 to cost. However, we do not want to know what an extra 13 cost, but what an extra one costs. To find it, we divide the \$10.00 into 13 equal parts, and call the result the extra cost of producing one more. In other words, to get the marginal cost of output in column 7, we need to divide the marginal resource cost by the marginal product.

Comparing columns 6 and 7, one can immediately see that 31 is the profit-maximizing level of output. Producing more or less will reduce profits. If the firm produces more than 31, it adds \$2.00 in revenue for each extra unit, but it also adds \$2.50 in cost. If it produces less, it cuts costs, but not by as much as it cuts revenues.

Checking the production function at 31 units of output, one sees that one needs four units of labor. Hence, finding profit-maximizing output and profit-maximizing input are two different ways to arrange the information from the production function, the supply of resources, and the demand for output. Both ways give identical results.

But you may ask what do real firms do?    