Get the production output equation. If your business has multiple plants, the production equation is likely to be different for each plant. For example purposes, assume that the production function is defined algebraically as Q = 10L^2 + 5L + 400, where "Q" and "L" represent production output and input labor costs, respectively.
Find the marginal product of labor. Differentiate the production equation with respect to labor. In the example, using rules of differential calculus, the marginal product of labor is 20L + 5.
Compute the marginal revenue at a given output price per unit. This is simply equal to the product of the marginal product of labor and the unit price. In the example, if the output price is $2 per unit, the marginal revenue product of labor is 40L + 10, or 2 times (20L + 5).
Calculate the optimal output. Set the marginal revenue product of labor equal to the marginal cost of labor to find the optimal level of labor input. Then substitute the result into the production equation to calculate the optimal output. In the example, assuming the marginal cost of labor is constant at $50 per hour, set the marginal revenue product of labor equation to this cost. So 40L + 10 = 50, which means the optimal level of labor input equals 1: (50 - 10) / 40. To conclude the example, substitute this value into the production equation, Q = 10L^2 + 5L + 400, to get Q = 10(1)^2 + 5(1) + 400 = 415. Therefore, the optimal output is 415 units.