This Demonstration illustrates the origin of the labor demand curve. A firm facing a fixed amount of capital has a logarithmic production function in which output is a function of the number of workers . The marginal product of labor (MPN) is the amount of additional output generated by each additional worker. In other words, MPN is the derivative of the production function with respect to number of workers, .
The tangent lines at certain points of the production function show that the MPN for any value of is simply the slope of a line tangent to the production function for that value of . The firm's profit-maximizing condition is when wage equals MPN (explained below), such that for any number of workers, the wage the firm is willing to pay is equal to the MPN associated with that value of . For this reason, the labor demand curve is simply the MPN. As productivity increases or decreases, MPN and therefore the labor demand curve respond by shifting to the right for a productivity increase and the left for a productivity decrease.
If the firm has a fixed amount of capital, its only variable input is labor. The marginal cost (MC) of labor is the wage and the marginal revenue (MR) for labor is the MPN, because that is the amount the firm receives per unit of labor. We know that all firms maximize profit when MC = MR, or in this case when the wage = MPN. Given these conditions, the labor demand curve is simply all of these points of MPN such that for any , the firm is willing to pay the corresponding MPN as a wage because that is how the firm maximizes profit at that value. As productivity increases, MPN increases, thereby shifting the labor demand curve out, which leads to higher wages.