Principles of Economics/Labor Production
Marginal product of labor
In firms, people start off rather productive, but as more people are added, specialization occurs, with the result that the marginal product of labor (the next worker's labor productivity) increases. After the critical point at which this has reached its maximum, the MPL begins to decline as additional workers become less productive. In the short run this is most acute, where a constant amount of available capital limits productivity. After a certain point, adding workers begins to reduce overall (total) productivity; this is reflected by a negative MPL.
Firms' production frontier
The gold line is the typical firm's production frontier. It is the sum of each worker's marginal productivity (taking the integral of MPL from 0 to L). At first the production frontier increases faster, then it increases slower, then it stops increasing, before beginning a decline (which should never happen if the firm is money-conscious).
Isoprofit lines set
Isoprofit lines are lines composed of amounts of labor and its total output that yield the same amount of profit for the firm. There are an infinite number of isoprofit lines, one above another, similarly to how there are an infinite number of indifference curves. The isoprofit line that is tangent to the production frontier is also the highest isoprofit line that the firm can reach.
Isoprofit lines part 2
There are a variety of isoprofit lines that may be drawn. Realistically, their slopes will be flatter than the ones shown here, and their y axis will be higher.
The slope is equal to wage divided by price of output.
The y intercept is profit divided by price of output.