We can break that question down. Profit is the difference of revenue minus cost. Ask, "What does one additional labor unit add to cost? What does one additional labor unit add to revenue?
The first question is relatively easy. What one additional labor unit will add to cost is the wage paid to recruit the one additional unit.
The second question is a little trickier. It's easier to answer a related question: "What does one additional labor unit add to production?" By definition, that's the marginal product -- the marginal product of labor is defined as the additional output as a result of increasing the labor input by one unit. But we need a measurement that is comparable with revenues and profits, that is, a measurement in money terms. Since the price is given, the measurement we need is the Value of the Marginal Product:
To review, we have made some progress toward answering the original question. Adding one more unit to the labor input, we have
| increase in revenue = | value of marginal product |
| increase in cost = | wage |
So the answer to "What will one additional labor unit add to profits?" is "the difference of the Value of the Marginal Product Minus the wage." Conversely, the answer to "What will the elimination of one labor unit add to profits?" is "the wage minus the Value of Marginal Product of Labor." And in either case the "addition to profits" may be a negative number: either building up the work force or cutting it down can drag down profits rather than increasing them.
So, again taking the bug's-eye view, we ask "Is the Value of the Marginal Product greater than the wage, or less?" If greater, we increase the labor input, knowing that by doing so we increase profits by the difference, VMP-wage. If less, we cut the labor input, knowing that by doing so we increase profits by the difference, wage-VMP. And we continue doing this until the answer is "Neither." Then we know there is no further scope to increase profits by changing the labor input -- we have arrived at maximum profits.
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