In economics, the labor demand of an employer is the number of labor-hours that the employer is willing to hire based on the various exogenous (externally determined) variables it is faced with, such as the wage rate, the unit cost of capital, the market-determined selling price of its output, etc.
The long-run labor demand function of a competitive firm is determined by the following profit maximization problem: where p is the exogenous selling price of the produced output, Q is the chosen quantity of output to be produced per month, w is the hourly wage rate paid to a worker, L is the number of labor hours hired (the quantity of labor demanded) per month, r is the cost of using a machine (capital) for an hour (the "rental rate"), K is the number of hours of machinery used (the quantity of capital demanded) per month, and f is the production function specifying the amount of output that can be produced using any of various combinations of quantities of labor and capital.
This optimization problem involves simultaneously choosing the levels of labor, capital, and output.
Depending on which effect predominates, labor demand could be either increasing or decreasing in r. The short-run labor demand function is the result of the same optimization except that capital usage K is exogenously given by past physical investment rather than being a choice variable.
The short-run labor demand function is derived the same way except with physical capital K being exogenous.