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Marginal Revenue Product |
Marginal Resource Cost |
Labor |
Output |
Marginal Product |
Marginal Revenue |
Marginal Cost |
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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?