Labor demand

Perfect competitor

The long-run labor demand function of a competitive firm is determined by the following profit maximization problem:

${\displaystyle \text{Maximize}pQ-wL-rK\text{with respect to}Q,L,\text{and}K}$

${\displaystyle \text{subject to}}$

${\displaystyle Q=f(L,K),}$

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. The resulting labor demand, capital demand, and output supply functions are of the general form

${\displaystyle L(p,w,r),}$

${\displaystyle K(p,w,r),}$

and

${\displaystyle Q(p,w,r).}$

Ordinarily labor demand will be an increasing function of the product's selling price p (since a higher p makes it worthwhile to produce more output and to hire additional units of input in order to do so), and a decreasing function of w (since more expensive labor makes it worthwhile to hire less labor and produce less output). The rental rate of capital, r, has two conflicting effects: more expensive capital induces the firm to substitute away from physical capital usage and into more labor usage, contingent on any particular level of output; but the higher capital cost also induces the firm to produce less output, requiring less usage of both inputs. 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.