We showed above that a profit-maximizing firm will hire the number of workers such that the wage is equal to the value of the marginal product of labor

But, as the text showed in an earlier chapter, a profit-maximizing firm will produce the quantity of output such that price equals marginal cost. Are these two rules inconsistent?

No. To see this let p = price of output, MPL = marginal product of labor, w = wage rate, VMPL = value of the marginal product of labor, and MC = marginal cost. To see that there is no inconsistency, note the following:
i. Firms maximize profits by producing the level of output such that price equals marginal cost and therefore p = MC.
ii. Firms maximize profits by hiring the number of workers such the wage rate equals the value of the marginal product of labor and therefore w = VMPL.
iii. By definition the value of the marginal product equals the price of output times the marginal product of labor and therefore w = p x MPL, which implies p = w / MPL
iv. MC must equal w / MPL. If a firm wants to produce one more unit of output it would need to hire 1 / MPL additional workers. So, for example, if the MPL is 2 then a firm would have to hire half an additional worker to produce one more unit of output. The cost of these additional workers is w / MPL, which is the cost of producing one more unit of output.
v. Therefore if a firm follows the w = VMPL rule in its hiring decision it will also follow the p = MC rule in its output decision.